Investigating a Possible Dynamical Origin of the Electroweak Scale

Transcription

1 Investgatng a Possble Dynamcal Orgn of the Electroweak Scale Unversty of Southern Denmark Master's Thess n Partcle Physcs Martn Rosenlyst Jørgensen CP 3 -Orgns Supervsed by Ass. Prof. Mads Toudal Frandsen, CP 3 -Orgns Postdoc Tomm Alanne, CP 3 -Orgns June, 7

2 Abstract One era came to an end n July when two experments, CMS and ATLAS, at the Large Hadron Collder at CERN announced the dscovery of a new resonance consstent wth the Standard Model Hggs boson. The Hggs boson was the last mssng pece of the Standard Model of elementary partcle physcs, our most fundamental descrpton of the elementary partcles and ther nteractons va three of the four forces, the excepton beng gravty. Although the Standard Model s n agreement wth a great number of expermental measurements, t cannot explan all the observatons. The thess begns wth an ntroducton to the Standard Model, the reasons why there must be some more fundamental theory beyond the Standard Model. Ths thess wll elucdate extensons of the Standard Model, where the Hggs sector s replaced by a strongly nteractng sector. We wll focus mostly on the naturalness.e. the problem that the mass of the Hggs boson s very ne-tuned. Therefore, n ths thess we wll nvestgate extensons of the Standard Model, where the standard Hggs sector s replaced by a strongly nteractng sector. After we have developed the tools to study strongly nteractng theores, we wll dscuss and develop three concrete examples: the Mnmal Walkng Techncolor (MWT) model, a Composte Hggs (CH) model and a Partally Composte Hggs (PCH) model. We wll nvestgate the vacuum stablty of the PCH model by calculatng the runnng of a new fundamental scalar self-couplng, and we dscover that ths knd of models are ne-tuned and the vacuum s unstable for a large part of the parameter space. Ths part of the thess s novel research.

3 Acknowledgements Ths master's thess s done at the Centre for Cosmology and Partcle Physcs Phenomenology (CP 3 - Orgns), Unversty of Southern Denmark. I would lke to thank my supervsors Mads Toudal Frandsen and Tomm Alanne for all the gudance. I would also lke to thank my fellow student Mette L. A. Krstensen who took tme to dscuss the thess wth me. Fnally, I would lke express my deepest grattude to Sophe and my famly for ther support, patence and love.

4 Contents Introducton 4 Introducton to Elementary Partcle Physcs 6. Symmetres Realzaton of symmetres Standard Model Untarty of W L W L Scatterng Ampltude Custodal Symmetry Custodal Symmetry at Tree Level Custodal Symmetry at Loop Level Trvalty and Vacuum Stablty Trvalty of QED Trvalty of Hggs Sector Vacuum Stablty n the SM Hggs Mass Correctons The EW Herarchy Problem Fne-Tunng of Models Fne-Tunng of the Hggs Mass Chral Symmetry Breakng n QCD Quantum Chromodynamcs (QCD) Constructon of an Eectve Lagrangan Chral Symmetry Breakng Techncolor Models Smple Techncolor Chapter Concluson Mnmal Walkng Techncolor The Underlyng Lagrangan for Mnmal Walkng Techncolor Low Energy Theory for MWT

5 CONTENTS 3.. Composte Scalars Composte Vector Bosons Fermons n the Eectve Theory Yukawa Interactons Extended Techncolor Models Walkng Techncolor Wenberg Sum Rules and the S Parameter Chapter Concluson Composte Hggs Dynamcs 8 4. The Fundamental Lagrangan Electroweak Vacuum Algnment The B Vacuum: The H Vacuum A Superposton of the two Vacua: Loop Induced Hggs Potental Gauge Contrbutons Top Contrbuton Explct Breakng of SU(4) Fne-Tunng of the Model Chapter Concluson Partally Composte Hggs Dynamcs The Fundamental Lagrangan Constructon of the Eectve Lagrangan The Vacuum Algnment Scalar Resonances The Normalzaton Factors The Angles n the Model The Parameter Space The Vacuum Stablty Chapter Concluson Conclusons 7 Appendces 4 Page 3 of 93

6 Chapter Introducton The Large Hadron Collder (LHC) s the bggest scentc nstrument ever created. It accelerates and colldes protons along the 7 klometers long tunnel excavated beneath the French-Swss border. The man physcs goal of the LHC s to determne the orgn of electroweak symmetry breakng,.e. the mehansm provdng the mass for the elementary partcles. The rst step towards ths goal was taken when the CMS and ATLAS collaboratons (Refs. [6, 7]) announced that they had dscovered a new resonance wth propertes consstent wth those of the Standard Model Hggs, wthn the measurement uncertantes. The Hggs boson was up untl the mssng pece of the Standard Model (SM) s responsble for the orgn of mass of the elementary partcles n that model, for curng the would-be volaton of untarty n the weak sector and to brng agreement between the predcted electroweak precson observables and the measured at Large ElectronPostron Collder (LEP) experments. The next goal s to measure and nvestgate what les beyond the SM. Despte of all successes of the SM t cannot explan all current observatons ncludng neutrno masses, baryogeness and dark matter, and there are a varous reasons that t s not the most fundamental theory of Nature. One mportant reason that the Standard Model may not be a complete theory of electroweak (EW) symmetry breakng s that the mass of the Hggs boson s very ne-tuned. The electroweak energy scale s namely 7 orders of magntude smaller than the Planck energy scale that characterzes gravty. It results n a naturalness problem of the electroweak scale whch s known as the electroweak herarchy problem, because t does not seem natural that the mass s extreme ne-tuned. In ths thess, our man motvaton s to search after a possble dynamcal orgn of the electroweak scale whch would be natural. Other ssues wth the SM Hggs sector are the trvalty problem and problems n avor physcs. These further motvate the quest for a theory of EW symmetry breakng beyond the SM Hggs model. The Naturalness paradgm wll be adressed n composte formulatons of the Hggs mechansm, ncludng so-called techncolor (TC) models, composte Hggs (CH) models, bosonc techncolor (BTC) models The rst step towards the Standard Model was the dscovery n 96 of a way to combne the electromagnetc and weak nteractons dscovered by Sheldon Glashow (Ref. [6]). In 967 Steven Wenberg and Abdus Salam ncorporated the Hggs mechansm (Refs. [596]) nto Glashow's electroweak nteracton gvng t ts modern form (Refs. [63, 64]). LEP collded electrons wth postrons at energes that reached 9 GeV (cf. Ref. [8]). In t was shut down to make way for the LHC, whch reused the LEP tunnel. 4

7 CHAPTER. INTRODUCTION and partally composte Hggs models (PCH). The man dea s to have technquarks and techngluons analogous to the quarks and gluons as n quantum chromodynamcs (QCD), that conne n technhadrons (technmeson and technbaryons) after chral symmetry breakng. Ths connement and chral symmetry breakng provdes a natural dynamcal orgn of the electroweak scale. By ntroducng techncolor the Hggs mechansm has a natural scale and s non-trval, but t stll does not explan the avor physcs. The TC models tself has no mechansm that explans the orgn of SM fermon masses. For that we would ntroduce extended techncolor (ETC). Such ETC models cause ther own set of problems. It s challengng to generate enough mass to the heavest fermons n some realzatons t s already problematc to produce the mass of the charm quark. Smultaneously, ETC contrbutes to the avor changng neutral currents (FCNC) and contrbutes to dscrepances wth precson electroweak measurements. The prmary soluton to these potental problems s to assume that the TC dynamcs s dstnctly unlke QCD. Ths scenaro s referred to as walkng techncolor (walkng TC), where the couplng constant of TC evolves slowly across a large energy scale as opposed to the 'runnng' couplng constant n QCD. Another ssue n these TC models s that t s heard to explan the mass of the observed 5 GeV boson at LHC. In TC the Hggs boson s dented wth the lghtest scalar resonance, the techn- (smlar to the resonance n QCD). By rescalng ths resonance n QCD to techncolor, t s too heavy to be the observed, unless the number of techncolors s very hgh (cf. Ref. []). Ths n turn s constraned by electroweak precson measurements. Ths ssue s allevated by CH and PCH (BTC n the TC lmt) models, where the Hggs boson s dented wth a composte Goldstone boson and a mxture of a composte Goldstone boson (scalar exctaton n the TC lmt) and a fundamental scalar, respectvely. Unfortunately, the parameters n both knd of models end up ne-tuned. By performng a novel computaton of the vacuum stablty n the PCH model n Ref. [3], we demonstrate that ts vacuum algnment angle seems ne-tuned. Ths thess conssts of a chapter that gves an ntroducton to the SM of elementary partcle physcs and ts problems, and three chapters that dscuss explct extensons of the SM Hggs sector rst presented n the three research papers n Refs. [3], respectvely. Ths thess s organzed as follows: In Chapter, the Standard Model, ts vacuum stablty, and ts problems mentoned above are dscussed, and how these ssues are addressed by a smple TC model whch s a rescaled QCD model. The Chapter begns wth a dscusson about the symmetres and why the symmetres along wth renormalzablty are the prmary reason for the predctve power of the Standard Model followed by a schematc revew of the Standard Model. The chapter ends wth a dscusson how the problems of EW symmetry breakng n the Standard model are addressed n a smple TC model whch s a scaled up verson of QCD. In Chapter 3, the mnmal walkng techncolor (MWT) model n Ref. [] s constructed, and we revew ETC and walkng TC. Chapter 4 ntroduces CH models (followng mostly Ref. []) by algnng the vacuum n another drecton away from the TC vacuum wth the motvaton to acheve a lght Hggs boson from a Goldstone Boson of the strong dynamcs. In Chapter 5, the potental ne-tunng problems n the CH models are addressed by ntroducng a PCH model as n Ref. [3], where the Hggs boson s partally composte and fundamental. We nally present a novel analyss of the vacuum stablty n ths model and the consequences for the vable parameter space of the model. Page 5 of 93

8 Chapter Introducton to Elementary Partcle Physcs In ths chapter, the Standard Model of partcle physcs and ts problems are dscussed. The chapter begns wth a dscusson of symmetres along wth renormalzablty and why the symmetres are the prmary reason for the predctve power of the Standard Model. Followng by a schematc revew of the Standard Model and a dscusson of each term n ts total Lagrangan. We ntate a dscusson of the possble problems of the Standard Model ncludng the untarty of W L W L scatterng, custodal symmetry, trvalty and vacuum stablty. By calculatng the mass correctons to the mass of the Hggs boson, we wll nd out that the Hggs mass s very ne-tuned, whch gves us a naturalness problem as dened and quanted by 't Hooft n Ref. []. The naturalness problem of the Hggs mass s called the electroweak (EW) herarchy problem, because the mass s 7 orders of magntude smaller than the Planck mass that characterzes gravty. At the end of the chapter, we ntroduce chral symmetry breakng n quantum chromodynamcs (QCD) and gves an ntroducton to a smple Techncolor model whch s motvated by tryng to address the naturalness problem of the Hggs mass.. Symmetres Let us start to dscuss the questons, what s a symmetry of a partcle physcs model, and what s the mportance of these symmetres n partcle physcs? A Symmetry means an nvarance under a set of transformatons. An popular example s a symmetrc geometrc object whch looks the same, f the object s rotated by an angle. The set of all symmetry transformatons form a symmetry group of the object. A rotaton s called a contnuous transformaton, whle for example a reecton transformaton of the object that keeps the object nvarant s called a dscrete transformaton. 6

9 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS The laws of Nature can be symmetrc. Ths means that the form of the equaton descrbng the law s mantaned under a change of space-tme coordnates and/or varables. We can categorze the symmetres such that geometrc symmetres act on space-tme coordnates and nternal symmetres do not. The contnuous symmetres of the equaton of motons can be related to conserved quanttes. Ths s quanted by the Noether's theorem (the precse mathematcal formulaton s gven n Ref. []), whch apples to both geometrc and nternal symmetres. Noether's Theorem: If the equatons of moton are nvarant under a contnuous transformaton wth n parameters, there exst n conserved quanttes. If the equatons of moton are nvarant under translaton n tme, translaton n space and rotatons n space, the correspondng conserved quanttes are shown n the three rst rows n Table.. In the case where we have relatvstc partcles, t s convenent to ntroduce Mnkowsk space M whch s a real fourdmensonal vector space wth the vectors x = (ct; ~x) and wth the metrc (ds) = dx dx = (dt) (d~x). A sem-drect product of the Lorentz transformatons x! x = x and the translatons n space-tme x! x = x + a (wth a M) form the Poncaré group, whch leave the Mnkowsk metrc nvarant. An elementary partcle should not depend on ts poston n space-tme or f the observer s n unform moton relatve to t. Therefore, the Lagrangan descrbng the partcle and ts nteractons should be nvarant under the Poncaré group. Contnuous Invarance Tme nvarance Translaton nvarance Rotatonal nvarance Gauge nvarance Conserved Quantty Energy Conservaton Momentum conservaton Angular momentum conservaton Charge conservaton Table.: Some symmetres and the assocated conservaton laws. We can wrte an nvarant Lagrangan under Poncaré transformatons L K ; (.) whch s the knetc term of a Drac fermon (x). Ths term s nvarant under a global U() phase transformaton (x)! exp(e) (x); (.) where e and are space-tme ndependent constants. If s space-tme dependent, Eq. (.) s no longer nvarant under the U() transformaton. The term can be nvarant by replacng the partal dervatve wth the covarant D + ea ; (.3) Page 7 of 93

10 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS where the gauge eld A transforms as A! (x): (.4) Ths procedure s called gaugng and t xes the form of nteractons between the Drac eld and the gauge eld A. The local phase transformatons are called gauge transformatons, and these knd of theores are called Abelan gauge theores. The local and also the global U() symmetry are contnuous symmetres and the correspondng conserved quanttes are the electrc charge of the Drac spnor (x) and the number of partcles, respectvely. Ths can be generalzed to non-abelan compact Le gauge groups. In ths case, all elds carry an addtonal ndex, or a, whch ndcates the charge wth respect to a gauge group for the fundamental and the adjont representaton, respectvely (Group representatons are dscussed n Appendx C). In these theores, a gauge eld can now be wrtten as A = A a TA a wth the non-commutng T A a generators n the adjont representaton of the gauge group, and A a are the component elds of the gauge eld for each charge. The gauge transformaton of a fermon eld s thus (x)! exp( a TA a ) (x) g (x); (.5) where a are arbtrary functons, and a takes the same values as for the gauge elds. The correspondng covarant dervatve s D + ea T (.6) wth the T generators n the fundamental representaton of the gauge group (n prncple they can also be n the adjont representaton nstead). The correspondng gauge transformaton for the gauge elds has then to take the nhomogeneous form n contrast to the Abelan theory n Eq. (.4) A! ga g + g@ g : (.7) The derence between an Abelan gauge and an non-abelan theores s that the generators of the gauge group are commutng and not commutng, respectvely. The gauge symmetry s an nternal symmetry, and the space-tme and the nternal symmetres are descrbed n terms of Le groups. Another type of symmetres than the Lorentz and the nternal symmetres are the dscrete symmetres. A dscrete symmetry s a symmetry that descrbes non-contnuous transformatons of a system. In addton to contnuous Lorentz transformaton, there are two other space-tme transformatons that can be symmetres of the Lagrangan: party and tme reversal. Party, whch s denoted by P, sends (t; ~x)! (t; ~x), whle the tme reversal, whch s denoted by T, sends (t; ~x)! ( t; ~x). At the same tme when we dscuss P and T, t wll be convenent to dscuss the dscrete transformaton: charge conjugaton, whch s denoted by C. Under ths transformaton, the partcles and antpartcles are nterchanged. Although any relatvstc eld theory must be nvarant under the Poncaré group, t need not be nvarant under the dscrete transformatons P, T and C. From experments, we know that the three of the four forces of Nature, the gravtatonal, electromagnetc, and strong nteractons, are symmetrc under P, T, and C. Page 8 of 93

11 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS The fourth force, the weak nteractons, volates both P and C, and certan rare processes n the Yukawa sector (processes nvolvng neutral K mesons) also show CP and T volaton. All experments ndcate that CP T s a symmetry of Nature... Realzaton of symmetres So far, we have consdered only exact symmetres. It s mportant to derentate what actually s symmetrc, Lagrangan or the vacuum, and at what scale the symmetry s manfest, and f t s broken, how t s broken. There s derent ways the symmetry can be broken. If the Lagrangan s nvarant under a symmetry for whch the vacuum s not nvarant ths symmetry s termed spontaneously broken. The symmetry can also be explctly broken va addng non-nvarant terms n the Lagrangan. It can not be excluded that the symmetry cannot be used to draw conclusons, f the breakng term s small. Some classcal symmetres of the Lagrangan can be spoled by the quantum eects, when we quantze the theory. Ths s called an anomalous symmetry, and the term that gves the breakng s called an anomaly. It s mportant for the consstency of the theory that all the local anomales are cancelled n the end, for example the gauge anomales n the Standard Model s cancelled as shown n Appendx B. The consequence by breakng symmetres s descrbed by the Goldstone theorem (derved at quantum level n Appendx D). The Goldstone theorem states the followng: If a subgroup H of the symmetry group G s broken, then there are dm(g=h) Goldstone bosons.. Standard Model In ths secton, we schematc summarze the Standard Model (SM) and brey dscuss each part of ts total Lagrangan. The SM s a SU(3)C SU()W U()Y gauge group. The three factors of the gauge symmetry gve rse to three fundamental nteractons (electromagnetc, weak nuclear and strong nuclear nteractons). The SM has been hugely successful n explanng expermental observatons, but t leaves some phenomena unexplaned. The SM does not ncorporate the full theory of gravtaton as descrbed classcally by the general relatvty, dark matter, dark energy, neutrno masses and oscllatons and baryogeness. Therefore, the SM s not a complete theory of the fundamental nteractons. The theory of the strong nuclear nteractons s a non-abelan gauge theory wth the gauge group SU(3)C. The quantum eld theory (QFT) of these nteractons s called the quantum chromodynamcs (QCD). Ths theory descrbes the nteractons between quarks and gluons, whch makes up hadrons (mesons and baryons) such as protons, neutrons and pons. The force carrers (the gauge bosons) n the theory are the gluons, and the assocate charge s called color (see Table.). The generators of QCD are the eght Gell-Mann matrces a. The theory of the electroweak nteracton s also a non-abelan theory wth the gauge group SU ()W U()Y. Ths gauge group s not smple, but t s a product of SU()W and U()Y, the uncaton of the electromagnetc wth the charged and neutral weak nteractons, whch s the combnaton by two Page 9 of 93

12 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS gauge couplng constants, g for the weak sospn SU()W and g for the weak hypercharge U()Y. The force carrers of the non-abelan gauge group SU()W are the massless W a bosons wth a = ; ; 3 and of the abelan gauge group U()Y s the massless B boson. The masses of the gauge bosons are ntroduced by spontaneous symmetry breakng, when the Hggs boson, = v + h, requres a non-zero vacuum expectaton value (vev), h = v, whch provdes three massve bosons called W and Z and one massless photon. Ths vev s also responsble for the fermon masses n the SM from the nteractons between fermons and the Hggs boson, whch are descrbed n the Lagrangan by the Yukawa terms. We have three generators of weak sospn called I a W = a = ( a are the Paul matrces) and the generator of weak hypercharge Y W. The EW gauge group spontaneously breaks to the electromagnetc symmetry group,.e. SU()W U()Y! U()Q, when the Hggs requres a vev. The QFT of ths gauge theory s called quantum electrodynamcs (QED). The generator of the electrc charge s dened va the Gell- Mann-Nshjma relaton (cf. Eq. (4..) n Ref. [3]) Q = I 3 W + Y W : (.8) The electrcal charges of the varous partcles n the SM are shown n Table.. SCALARS Symbol Name Electrc charge Baryon number Lepton number Gauge representatons Hggs doublet (,) (,,) FERMIONS Symbol Name Electrc charge Baryon number Lepton number Representaton Q L I Left-handed quark (/3,-/3) /3 (3,,/3) u R I Rght-handed up quark /3 /3 (3,,4/3) d R I Rght-handed down quark -/3 /3 (3,,-/3) L L I Left-handed lepton (,-) (,,-) e R I Rght-handed electron lepton - (,,) GAUGE FIELDS Symbol Assocate charge Electrc charge Group Couplng Gauge Gauge representatons B Weak hypercharge U()Y g' (,,) W ;;3 Weak sospn SU()W g (,3,) G Color SU(3)C g s (8,,) Table.: The content of elds n the SM. If we have a doublet (e.g. the Hggs doublet ) then ts each electrc charges are represented as (U()Q charge of rst component, U()Q charge of second component), e.g. (,) for the Hggs doublet. The representatons of the elds under the gauge groups SU (3)C, SU()W and U()Y are lsted as (SU(3)C, SU()W, U()Y). For example, the gluons have the gauge representatons (8; ; ), because there s a color octet of gluons, whch all are weak sospn snglets wth hypercharge zero. Page of 93

13 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS It s often convenent to denote the left-handed quarks and leptons doublets by Q L I ul I d L I A and L L I L I e L I A ; (.9) and the rght-handed fermon snglets by u R I, dr I, and er I. Here u; d;, and e represent up-type quark, down-type quark, neutrno, and electron-type lepton, respectvely, and I s the generaton ndex (I; J; K; = ; ; 3). The three derent generatons of the SM contan u I =fu; c; tg; d I = fd; s; bg; I = f e ; ; g; e I = fe; ; g: (.) The content of the elds n the SM and ther quantum numbers s shown n Table.. The representatons of the elds under the gauge groups SU(3)C, SU()W and U()Y are lsted as (SU(3)C, SU()W, U()Y) n the table. For example, the gluons form a color octet whch all are weak sospn snglets wth hypercharge zero,.e. they have the gauge representatons (8,,). Fnally, we wll formulate the Lagrangan of the SM. The Lagrangan of the SM must respect the gauge symmetres, Lorentz nvarance and renormalzaton. It s useful to dvde the total Lagrangan nto four parts as follows L SM = L G + L F + L H + L Y : (.) The rst term contans the Yang-Mlls terms for the gauge elds, whch reads where the gauge eld strength tensors are dened as L G = 4 W W 4 B B 4 Ga G a ; (.) W =@ W + g"jk W j W k ; B =@ B ; (.3) G a =@ G G a + g sf abc G b G c ; where = ; ; 3 and a = ; : : : ; 8. The structure constants are dened as [ a ; b ] = f abc c and [ ; j ] = " jk k, where a and are the generators of SU(3)C and SU()W gauge group, respectvely. The second term s the fermon terms, the knetc term and ther nteractons wth the gauge bosons, whch are X X L F = ( L L I =DL L I + Q L I =DQ L I ) + (e R I =De R I + ur I =Du R I + d R I =Dd R I ); (.4) where the covarant dervatve s I D g I W + g Y W B g s a Ga : (.5) The thrd part of the Lagrangan contans only the Hggs and the electroweak gauge bosons L H =(D ) y (D ) V ( ) =(D ) y (D ) + y ( y ) ; (.6) Page of 93

14 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS where the covarant dervatve s and the Hggs complex doublet s (x) = D + (x) (x) W + g Y W B ; (.7) A = (x) + (x) (x) 3 (x) A : (.8) We have that = ( )= p. The mass terms for the gauge bosons come from the knetc term after the Hggs boson has acqured a vev, whch s v = 46 GeV. Therefore, the physcal Hggs eld, h, s an exctaton around the vev, v, and we would wrte = v + h wth the expectaton value h = v. The physcal content of EW symmetry breakng can be extracted most easly n the untary gauge, where the would-be Goldstone boson components, ;;3, are set to zero (cf. page 58 n Ref. [3]). In ths gauge, there are no unphyscal elds and we can classfy the physcal elds as egenstates of electrc charge and mass. In ths gauge, the Hggs doublet s thus (x) = v + h A : (.9) A mass term of the form m W a W a for the gauge bosons s not nvarant under non-abelan SU()W gauge transformatons n Eq. (.7), and therefore t s forbdden. The spontaneous electroweak (EW) symmetry breakng of the followng terms n the knetc term of Eq. (.6) gves g 4 y W a W a + g g 4 y B B + 4 y (gw 3 + g B )(gw 3 + g B ) SB! W a W a (v + h) + g B B (v + h) + (gw 3 + g B ) (v + h) = g v W + W + (g + g )v Z Z + + g v hw + W + (g + g )v hz Z g hhw + W + g + g hhz Z : 4 8 We have that the mass egenstates of the gauge bosons are W (x) = A Z A c W W (x) W (x) and s W A A s W c B W 3 (.) (.) wth the weak mxng angle (the Wenberg angle) c W cos W = g g p and s W sn W = p g + g g + g ; (.) whch rotates the orgnal W 3 and B vector boson plane. Ths rotaton gves one postvely and negatvely charged gauge boson, W bosons, two neutral gauge boson, Z boson and the photon A. Accordng to Eq. (.), the masses at tree level of these gauge bosons are Page of 93

15 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS p m gv g W = ; m + g v Z = The Hggs couplngs to the massve gauge bosons are = m W c W ; m = : (.3) ghww SM = g v ; gsm hzz = (g + g )v ; ghhww SM = g ; 4 gsm hzz = g + g : 8 (.4) In the future, we would dene I as a Drac spnor n the generaton I; J; K; = ; ; 3, whch can be ether u I, d I, I or e I. The left- and the rght-handed can be projected out wth the projecton operators P L;R = ( 5 )= as follows L;R = P L;R. So far all the fermons are massless. A Drac mass term s not allowed, because the SU()L symmetry transforms the eld e L nto another eld L. Under such a transformaton the mass term (cf. Eq. (7.67) n Appendx C-) m = m( L R + R L ) (.5) s clearly not nvarant, and therefore t s forbdden. Agan, we can generate a mass term va the Hggs mechansm. We can construct a term that s a product of the Hggs and one of the SU ()L doublets of the left-handed fermons as n Eq. (.9). These terms are called Yukawa nteracton terms, and the Yukawa Lagrangan n the SM s L Y = Q L I G u IJu R J c Q L I G d IJd R J L L I G e IJe R J + h.c. (.6) where G e, G u and G d are 3 3 matrces, and the fermon elds L;R I are the charge egenstates of the weak nteracton. The eld c(x) s the charge-conjugate Hggs eld c(x) = (x) = ( ; (x)) (cf. page 595 n Ref. [3]), where a are the Paul matrces n Eq. (7.3) n Appendx A. It follows that the conjugated Hggs eld c(x) also transforms as a SU() doublet, because the dentty exp( a a =) = exp( a a =) : (.7) After the spontaneous symmetry breakng ( = v + h), we have the terms G IJ ( L;I R;J + h.c.) SB! v p G IJ ( L;I R;J + h.c.) = M IJ ( L;I R;J + h.c.); (.8) where the mass matrces for up-type quarks, down-type quarks, and electron-type leptons are M u IJ = v p G u IJ; M d IJ = v p G d IJ; and M e IJ = v p G e IJ: (.9) These mass matrces can be dagnoalzed by a b-untary transformaton for left-handed and rght-handed fermons, respectvely, resultng n the fermon mass egenstates, Thus, the fermon masses are L I = X K U ;L IK L K and R I = X K U ;R IK R K: (.3) Page 3 of 93

16 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS m ;I = p X K;M U ;L IK G KM U ;Ry MI v: (.3) Thus, the fermons are no longer n the charge egenstates of weak nteracton, but they are mass egenstates. Thus, the rst term n the Yukawa Lagrangan n Eq. (.6) can be wrtten as Q L I G u IJu R J c = X I and the second term s X Q L I G d IJd R J = u L I V q IJ I;J and the thrd term s m u;i X v u I ( + 5 )u I (v + h 3 ) + I;J p mu;j X L L I G e IJe R J = I L VIJ l I;J v p me;j v X d R J + + I X e R J + + d L I V qy IJ p mu;j v u R J ; (.3) m d;i v d I ( + 5 )d I (v + h + 3 ); (.33) I m e;i v e I ( + 5 )e I (v + h + 3 ); (.34) whch are derved n Eqs. (7.68)-(7.7) n Appendx C-. We have used Eq. (.3), Eq. (.3) and the antcommutaton relaton f 5 g = to rewrte these Yukawa terms. The neutral currents whch are not changng avors, the combnatons U ;L (U ;L ) y = always appear, and they are not aected. For the avor-changng currents we have the matrces V q =U u;l (U d;l ) y ; V l =U ;L (U e;l ) y ; (.35) whch provdng the avor mxng. The matrx V q s the CKM matrx for quark mxng, and the matrx V l s the PMNS matrx for possble lepton mxng. By nsertng the Yukawa terms n Eqs. (.3)-(.34) nto Eq. (.6), we obtan that the total Yukawa Lagrangan n the SM can be wrtten n terms of Drac spnors as follows L Y = Q L X = I G u IJu R J c Q L I G d IJd R J L L I G e IJe R X f =u;d;e I X p + I;J X I;J m u;i v p v m e;j m f f I f I X f =u;d;e X I J + h.c. m f;i v ( f I f I h Iw;f 3 f I 5 f I 3 ) u R I V q IJ dl J + + d L I V qy IJ ur J m d;j u L I V q IJ dr J + + d R I V qy IJ ul J L I V l IJe R J + + e R I V ly IJ L J ; (.36) where we have used Eq. (7.7) n Appendx C-. Thus, the four Lagrangan parts n the total Lagrangan of the SM n Eq. (.) are gven n Eq. (.), Eq. (.4), Eq. (.6) and Eq. (.36), respectvely. In the followng secton, we wll nvestgate how the Hggs boson untarzes the SM scatterng ampltudes. We wll examne the untarty of W L W L scatterng ampltude. In ths dscusson we wll dscover that we need a scalar partcle to untarze ths scatterng process, because the scatterng ampltude grows wth the energy s=m W wthout the scalar, where s s the center-of-mass (CM) energy squared of the Page 4 of 93

17 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS W L W L scatterng. As we wll see, the current data on the observed Hggs couplng to W bosons stll allow room for addtonal doublets besdes the dscovered Hggs boson. Therefore, t remans an open queston, whether the dscovered Hggs boson s the only one responsble for the full EW symmetry breakng..3 Untarty of WLWL Scatterng Ampltude A new scalar partcle wth mass of approxmately 5 GeV was dscovered at the Large Hadron Collder (LHC) n July. Ths s consstent wth beng the long sought Hggs boson of the SM, whch was proposed n 96s (n Refs. [596]). Ths Hggs partcle cures would-be volaton of untarty of the scatterng ampltudes n the SM. In ths secton, we wll show that scatterng of the longtudnal components of the weak gauge bosons s a useful probe of EW symmetry breakng. The SM scatterng ampltudes (e.g. the ampltudes of the dagrams n Fgure.) n the SM wthout Hggs exchange grow wth the energy as s=m W, where s s the center-of-mass (CM) energy squared of the W LW L scatterng. I.e. the ampltude of the W L W L scatterng dverges wth the energy, and thus t s not untary. Includng the Hggs boson exchanges as shown n Fgure., total W L W L scatterng ampltude s untarzed. W + L; (p ) W L; + (q ) W L; + (p ) W L; + (q ) W L; + (p ) W + L; (q ) Z; p + p p q Z; W L; (p ) W L; (q ) W L; (p ) W L; (q ) W L; (p ) W L; (q ) Fgure.: The dagrams that contrbute to the ampltude of the W L W L scatterng wth purely weak gauge bosons contrbutons. W + L; (p ) W + L; (q ) W + L; (p ) W + L; (q ) h p + p p q h W L; (p ) W L; (q ) W L; (p ) W L; (q ) Fgure.: The dagrams that contrbute to the ampltude of the W L W L scatterng wth the Hggs boson contrbutons. Now, we consder the process W + (p )W (p )! W + (q )W (q ), whch gets contrbutons from the Feynman dagrams of a four-pont vertex and ; Z n both s and t channels, as well as the dagrams wth a Hggs propagator n both s and t channels. The ampltudes for the gauge dagrams n Fgure. can be wrtten as Page 5 of 93

18 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS M 4 (W L ; W L! W L ; W L ) = e [ s L (p ) L (q ) L (p ) L (q ) L (p ) L (p ) L (q ) L (q ) W M Z; s M Z; t L (p ) L (q ) L (p ) L (q )] (W L ; W L! W L ; W L ) = e s + c w=s W s m Z p L (p ) L (p ) + q L (q ) L; (q ) (W L ; W L! W L ; W L ) = e t + c w=s W t m Z p L (q ) L (p ) p L (q ) L; (p ) h (p p ) L (p ) L (p ) + p L (p ) L (p ) h (q q ) L (q ) L (q ) q L (q ) L; (q ) h (p + q ) L (p ) L (q ) q L (p ) L (q ) h (p + q ) L (p ) L (q ) q L (p ) L; (q ) ; where L (k) s the longtudnal polarzaton four-vectors of the W bosons wth momentum k. The ampltudes for the Hggs boson dagram n Fgure. are gven by M Hggs s (W L ; W L! W L ; W L ) = L (p ) em W M Hggs t (W L ; W L! W L ; W L ) = L (q ) em W g L s (p ) W (p + p ) m h g L s (p ) W (p q ) m h L (q ) em W s W g L (q ) L (q ) em W s W g L (p ): (.37) To calculatng these ampltudes we need the longtudnal polarzaton four-vectors L, whch are derved n Appendx C- n Eqs. (7.75)-(7.84). In the center-of-mass frame of the ncomng W + (p )W (p ) par where ~p = ~p, accordng to Eq. (7.83) and Eq. (7.84) we can express the longtudnal polarzaton four-vector as and smlarly where s = (p + p ) L (p ) = p m W L (p ) = p m W m W s m W s p ; (.38) p ; (.39) = 4 (p ). For the outgong W + L (q )W L (q ) par ther longtudnal polarzaton vectors can be obtaned by smply make the substtuton (p ; p )! (q ; q ). However, we need to wrte the varous products between the four-momentum vectors n terms of the Mandelstam varables, s, t and u, n Eq. (7.7) as n Appendx C-, whch are gven n Eq. (7.73). Fnally, we need also the relaton where the sum of Mandelstam varables gves s + t + u = 4m W. These longtudnal polarzaton vectors and these expressons can be substtuted nto the above ampltudes, to leadng term of order O(E 4 =m 4 W ) of each ampltude we have calculated them n Eqs. (7.85)- (7.95) n Appendx C- to be M 4 (W L ; W L! W L ; W L ) = e 4m 4 W s W M Z; s (W L ; W L! W L ; W L ) = e 4m 4 W s W s + 4st + t 4m W (s + t) 8m W s s(t u) 3m W (t u) ut + O E m W ; + O E m W ; Page 6 of 93

19 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS M Z; t (W L ; W L! W L ; W L ) = e t(s 4m 4 u) 3m W s W (s u) + 8m W W s M Hggs s (W L ; W L! W L ; W L ) = e 4s W m W M Hggs t (W L ; W L! W L ; W L ) = e 4s W m W s m W s m h t m W t m h u + O E m W ; + O E m W ; (.4) + O E m W : The sum of the gauge dagrams n Fgure. (cf. Eq. (7.74 n Appendx C-) s M Gauge (W L ; W L! W L ; W L ) = M 4 + M Z; s + M Z; t = e u 4s + O E W m4 m W : (.4) W The gauge structure ensures the cancellaton of the O(E 4 =m 4 W ) terms. The problem s that the sum of the gauge dagrams are left wth O(E =m W ). Therefore, for the scatterng ampltudes wth purely gauge bosons wthout Hggs bosons, the ampltudes grow wth the energy as s=m W, and thus t s not untarzed. However, we have the contrbutons from the Hggs dagrams n Fgure., whch are M Hggs (W L ; W L! W L ; W L ) = M Hggs s ' e 4s W m W e = 4s W m W + M Hggs t = e 4s W m W " s m W s m h (s + t) + O E m W = e 4s W m W u + O E m W t + m W t m h # + O E m W 4m W u + O E m W (.4) n the lmt s m h ; m W. Totally, the W LW L scatterng ampltude s M T otal = M Hggs + M Gauge = O E m W : (.43) Therefore, n the SM wth the Hggs boson the ampltude s completely untarzed by the Hggs boson. Once p s goes beyond the Hggs boson mass, then the scatterng ampltude wll no longer grow lke s=m W. However, because current data stll contans sgncant uncertantes. There s stll room for a non-sm Hggs sector, e.g contnang more doublets lke n the HDM. One mportant measurment s the constrants of the Hggs couplng to W and Z bosons par n Eq. (.4),.e. g SM hww = g v= and g SM hzz = (g + g )v=, respectvely. The current data for the rato of ths couplng measured for the combnaton of the ATLAS and CMS measurements and n the SM (cf. table 8 n Ref. [36]) s W g hww ghww SM = :9 +: : (.44) wth CL ntervals. The central value s close to one,.e. that the observed Hggs boson leaves lttle space for the exstence of another Hggs boson or some physcs beyond the SM whch couples to the W bosons. The Mandelstam varable s s related to the other two Mandelstam varables, e.g. u, wth the relaton s + t + u = 4m W. Page 7 of 93

20 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS On the other hand, f the Hggs boson couplng to the W bosons devates from the SM value, then the ampltude for the Hggs dagrams n Fgure. s moded wth the rato W n Eq. (.44) squared from the two hw W vertces n the dagrams. The sum of the moded Hggs ampltudes are M Hggs (W L ; W L! W L ; W L ) = W e 4s W m W u + O E m W : (.45) Even for small devaton from the SM value, the terms grow lke u=m W (related to s=m W ) n the scatterng ampltude, and these terms would blow up after httng the mass pole of the Hggs boson. Thus, accordng to Eq. (.44) there s a possblty for a new Hggs doublet or other physcs beyond the SM whch s responsble for untarty of the scatterng ampltude. The mportance of ths secton s that we need a scalar or somethng else to untarze the W L W L scatterng ampltude, where the Hggs boson s responsble for t n the rst place. The next mportant feature dscussed n next secton s the custodal symmetry. In the SM, there s a global symmetry of the Hggs potental n the SM (n Eq. (.6)). In ths secton, we wll gve the constrants on the sze of the break of the custodal symmetry measured by the LEP experments, whch gves only small room for models beyond the SM that breaks the custodal symmetry..4 Custodal Symmetry One aspect we have glossed over so far s the necessty of two doublets for the Hggs. When we have only one doublet, then n ths case the number of Goldstone bosons s too small to provde three massve gauge bosons whch s demanded by experment. Thus, the presence of two doublets s expermentally necessary. The extra doublet has more consequences than just gvng all three weak gauge bosons mass..4. Custodal Symmetry at Tree Level To understand these consequences t s best to concentrate rstly on the pure Hggs part of the Lagrangan, L H, n Eq. (.6), whch reads L H = (D ) y (D ) V ( ) = (D ) y (D ) y ( y ) ; (.46) where the Hggs doublet s (x) = (x) + (x) (x) 3 (x) A : (.47) Rewrtng the Hggs doublet n Eq. (.47) wth four degrees of freedom to a matrx M (x) p ((x) + (x)) = ( c(x); (x)) = (x) + 3(x) (x) + (x) Now, the pure Hggs Lagrangan n Eq. (.46) can be rewrtten to L H = Tr[D M (D M ) y ] Page 8 of 93 (x) + (x) (x) 3 (x) A ; (.48) Tr[MMy ] 4 Tr[MMy ] ; (.49)

21 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS where the covarant dervatve s D M M gw a a M + g MB 3 : (.5) Ths Lagrangan s dentcal to the Lagrangan n Eq. (.46). Ths can be seen by multplyng two doublets together, whch gves y = Tr[MMy ] = and ; (.5) and by wrtng out for example the rst term wthout dervatve n the knetc term, whch gves y g W g W = g 8 [ ]W 3 W and y Tr gw M gw M = g 8 [ ]W 3 W : (.5) Ths can also be shown for the rest of the terms n the knetc term. By nsertng these expressons above nto Eq. (.49), we obtan the Hggs Lagrangan n Eq. (.46) agan. The next thng, we do, s to set the EW couplng constants g; g =, then we have the global symmetry group SU()L SU()R, whch s somorphc to SO(4) (.e. SU()L SU()R = SO(4)). By rewrtng the Lagrangan wth the doublet to one wth the M matrx, we can now see the SU()L and SU()R symmetres. We have namely that the Lagrangan wth the M matrx n Eq. (.49) s nvarant under the global transformaton where g L SU()L and g R SU()R, because M! g L Mg y R ; (.53) Tr[MM y ]! Tr[g L Mg y R g RM y g y L ] = Tr[g LMM y g y L ] = Tr[MMy ]; (.54) and therefore the knetc term s also nvarant under these global transformatons. Thus, we have rewrtten the Hggs Lagrangan n a form such that we can see both the SU()L and the SU()R symmetry. When we take the mass parameter to be negatve and the self-couplng postve, then at tree level we obtan h v = j j 6= and = v + h; (.55) where h s the Hggs eld and v s vev of the Hggs eld. The global symmetry group SO(4) wll break to SO(3) whch s somorphc to SU()V,.e. that SU()L SU()R! SU()V, when one of the degrees of freedom gets xed, because the expectaton value n one drecton s derent from zero, h y = v =. The symmetry group SU()L SU()R breaks to SU()V, because h = htr(m )! Tr(g L Mg y R ) = htr(g y R g LM ) = htr(m ); (.56) f and only f g L = g R = g V. I.e. that the global group breaks to SU()V, when the Hggs has a vacuum expectaton value. Page 9 of 93

22 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS Now, we are gaugng the weak sospn SU()L and the hypercharge U()Y gauge group by settng the couplng constants g 6= and g 6= n the covarant dervatve n Eq. (.5). More precsely the SU()R symmetry s broken explctly, because the U()Y subgroup of t s gauged. Ths can be seen n the followng term n the knetc term of the Lagrangan n Eq. (.49) by transformng t, h Tr gw a a M g MB 3 = gg W a B Tr[ a M 3 M y ]! 8 gg W a B Tr[ a g L Mg y R 8 3 g R M y g y gg L ] 6= W a B Tr[ a M 3 M y ]; 8 (.57) whch s not nvarant under the global transformaton M! g L Mg y R of the symmetry group SU() L SU()R. Therefore, the hypercharge gauge group U()Y breaks the global symmetres down to a subgroup SU()W U()Y, when t s gauged (.e. when g 6= ). Thus, we have that the M matrx transforms now globally as M! g W Mg y Y to keep the Lagrangan nvarant, where g W SU()W and g Y U()Y. Overall, when we are gaugng some of the symmetry groups, then we break some of the global symmetres. Therefore, we have now the followng symmetry breakng pattern for the EW symmetry breakng SU()W U()Y! U()Q; (.58) where U()Q s the electromagnetc gauge symmetry. Ths gves the three massless Goldstone bosons,.e. (= p )( ) and 3, accordng to the Nambu-Goldstone's theorem. These Goldstone bosons become the longtudnal degree of freedom of the massve electroweak gauge bosons W and Z n the untary gauge. If the couplng constants are g 6= and g =, then we have stll the symmetry breakng pattern SU()L SU()R! SU()V. In ths case, the covarant dervatve s D M M gw a a M: (.59) By nsertng = v + h nto the knetc term n the Lagrangan n Eq. (.49), we obtan the mass term of the W bosons a Tr[D M (D M ) y ] M gw a =Tr hg W a a b MMy =Tr = g v h g v M y + gw a M y a W b + = Tr 8 W a ab W b + Tr 4 W a W a + : : : : Accordng to Eq. (.) we have for g = that and therefore we get that h g v h g 4 W a a v b W b + : : : 8 W a " abc c W b + : : : (.6) W + = W W p ; W = W + W p and Z = W 3 ; (.6) W + W = (W W + W W ) and Z Z = W 3 W 3 : (.6) Page of 93

23 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS By nsertng these expressons nto Eq. (.6), we obtan the mass terms of W and Z Tr[D M (D M ) y ] = g v = g v 4 W a W a + = g v 4 (W + W + Z Z ) + : : : ; 4 (W W + W W + W 3 W 3 ) + : : : (.63).e. that the masses of the W and Z bosons are degenerated, whch are m W = m Z = gv=. Ths degeneraton s because of the custodal symmetry whch s the global symmetry group SU ()V. In ths case, the Wenberg angle n Eq. (.) s cos W = g p g + g = : (.64) Therefore, we have the followng relaton between the masses and the Wenberg angle m Z m W cos W = : (.65) On the other hand, f both couplng constants are derent from zero, g 6= and g 6=, then we have the symmetry breakng pattern SU()W U()Y! U()Q. In that case, the covarant dervatve of M s D M M gw a a M + g M 3 B : (.66) When the Hggs eld requres a vacuum expectaton value h = v, then we obtan the followng terms from the knetc term Tr[D M (D M ) y ] =Tr hg W a a b MMy W b + Tr hg M 3 B 3 Tr hgg W 3 3 M 3 M y B Tr hgg M 3 B WM 3 y 3 = g v Accordng to Eq. (.), we have 4 W a W a + g v B B gg v 4 4 M y B W 3 B + : : : = v 4 (g W + W + g W 3 W 3 + g B B gg W 3 B ) + : : : : (.67) Z Z = g + g ( g B + gw 3 )( g B + gw 3 ) = g + g (g B B + g WW 3 3 gg WB 3 ): (.68) Therefore, we get that the mass terms for the W and Z bosons are Tr[D M (D M ) y ] = g v W + W + v 4 4 (g + g )Z Z : (.69) The custodal symmetry s now broken, and therefore the masses of the gauge bosons are no longer degenerate: m gv W = and m v Z = p g + g : (.7) The relaton between the masses of the W bosons and the Z boson at tree level can be wrtten as Page of 93

24 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS = m W cos W m Z = cos W = cos w (v=) gv= p g + g = g cos W p g + g ) (.7) g p g + g ; Thus, the custodal symmetry s hdden n the cosne of the Wenberg angle cos W. We wll see n the followng that the Yukawa sector breaks the custodal symmetry..4. Custodal Symmetry at Loop Level The Yukawa nteractons do not respect the custodal symmetry. At loop level, there s a very small addtonal contrbuton to the left-hand sde of the relaton n Eq. (.7). We dene that the rght-hand sde s equal to some parameter, such that we obtan the mass relaton The Lagrangan wth the Yukawa nteractons for the quarks n Eq. m W m Z cos W + : (.7) rewrtten n terms of Weyl spnors, whch for one generaton has the form (.36) wthout avor mxng s L Y = u " j q Lj u R d q y L d R + h.c. = u " j q Lj u R d q y L d R u " j q y L yj u R d q L y d R; (.73) where the Yukawa couplngs are q = p m q =v and ; j; = ; are SU()W ndces. By usng the M matrx n Eq. (.48) the Yukawa terms can be rewrtten to q y L u R whch s nvarant under the custodal symmetry q y L u R d R A + h.c.! q y L gy L g LMg y R g R d R A + h.c.; u R d R A + h.c. = q y L u R d R A + h.c.: (.75) The problem s that ths s for the case where the Yukawa couplng constants, u and d, are the same, but ths s not the case n the SM. The Yukawa terms can nstead be rewrtten to where L Y = u q y L MP U P U u R d A d q y L MP u R d R A and PD These terms are not nvarant under custodal transformatons, L Y! u q y L gy L g LMg y R P U g u R d R A d q y L gy L g LMg y R P Dg u R A + h.c.; (.76) A : (.77) d R A + h.c. = Page of 93

25 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS u q y L Mgy R P U g u R d R A d q y L Mgy R P Dg u R d R A + h.c. 6= L Y ; because P U or P D do not commute wth g R. Therefore, the Yukawa terms n the SM are not custodal symmetrc, f the masses of the fermons n each generaton are derent. It wll gve rse to contrbutons to the parameter from the loop dagrams, whch lead to the mass correctons to the masses of the W and Z bosons. In Fgure.3 are shown the one-loop dagrams that lead to mass correctons of the two masses wth the two heavest fermons, the top and the bottom quark. The Yukawa couplng s expressed n the fermon propagators n the loops. These knds of correctons to the parameter (cf. Eq. () n Ref. [44]) are 3v f = 6m W s W mf v mf ' :8 : (.78) v where the electromagnetc ne-structure constant (m Z ) = 7:95 :7 (Ref. [44]), m W = 8:48 :39 GeV (Ref. [73]) and s W = :36 :4 (Ref. [73]). Ths correcton s very small even for the heavest fermons wth masses m t ' 7 GeV and m b ' 4 GeV compared to the vev of the Hggs boson v = 46 GeV. Ths gves t = :88 for the top-loop correcton. In Appendx I, the so-called T parameter s dened n terms of the self-energy of the vector bosons, whch s one of the EW precson parameters measured at the LEP experments. The T parameter s normalzed to be zero n the SM. Data from the LEP experments constran the T parameter to be T = :8 : (cf. Eq. (.7) n Ref. [73]). It s related to the parameter (cf. Eq. (.68) n Ref. [73]) as follows = (m Z )T ' + (m Z)T: (.79) Therefore, the expermental measurement of the parameter from the LEP experments s = :6 :9, where the loop correctons n the SM are ncluded such that the parameter s normalzed to be one n the SM. t; b t; b W W Z Z Fgure.3: The loop dagrams whch gve rse to mass correctons of the W and Z bosons and further provde a correcton to the parameter. If we consder an extra fermon doublet (U; D) wth the usual left-handed couplng to SU()W, hypercharge Y and masses m U and m D, then t contrbutes to the T parameter (cf. Eq. (4.) n Ref. [4]) Parameters n QFTs depend on whch energy scale they are measured, e.g. the mass (mz) s renormalzed at the Z boson mass. Ths s dscussed more clearly later, when we talks about the runnng of couplngs. Page 3 of 93

26 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS wth (m T U m D ) s W c W m Z (m U m D ) v ; (.8) where we assume that jm U m D j m U ; m D. Therefore, the ncrease of the T parameter by addng a fermon doublet depends on the squared rato of the mass splttng between the two doublet components and the EW vev. Thus, the thrd generaton of the quarks n the SM model contrbutes mostly to the T parameter. We can conclude, f the parameter s measured to be greater than the SM's predctons of t, then there should be somethng new physcs. Expermentally, the parameter s measured to be very close to one. If we want to extend the SM, then the extenson must only provde a very small contrbuton to the parameter. Therefore, new physcs should be custodal symmetrc or the symmetry must be broken mnmal. In the followng secton, we wll focus on the possble problems, called trvalty and vacuum stablty. Trvalty s a possble problem n QED and n a pure Hggs sector, whle t turns out that possbly the SM s vacuum unstable..5 Trvalty and Vacuum Stablty In (non-conformal) quantum eld theory a change of the renormalzaton group (RG) scale nduces a change n the couplng constants g of the theory. We say that the couplng constants run wth energy. The runnng of the couplng constants encodes mportant features of a theory, e.g. asymptotc freedom, trvalty, vacuum stablty and uncaton etc.. Let us examne two potental problems n the SM related to the runnng couplngs, whch are trvalty and the vacuum stablty problem. g g L Trval Theory g Asymptotc Theory g g Unstable Theory g E Fgure.4: Derent examples of how a couplng g can run and ts -functon for a trval, an asymptotc and an unstable theory. Page 4 of 93

27 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS The dependence of g() upon can be expressed n terms of a -functon of the theory (cf. Eq. (.6.4) n = [g()]; (.8) where n Appendx E t s shown how the -functon s derved for derent couplng constants to one-loop order. A trval theory means that the theory needs to be non-nteractng (the couplng s zero) to be a consstent theory. As we wll see there s a Landau pole problem n QED, because ts couplng constant blows up to nnty at a nte energy scale L as sketched n upper panel n Fgure.4, because ts -functon ncreases wth ncreasng couplng constant. QED s not the only theory wth a Landau pole problem. The scalar quartc couplng n the Hggs sector has the same problem. On the other hand, the QCD theory s a non-trval theory, because t s an asymptotc theory whch couplng s gong asymptotcally to zero at hgh energes as sketched n the mddle panel n Fgure.4. Fnally, we have that the vacuum of the theory can be unstable f ts couplng s gong to negatve values at energes above an nstablty energy E as sketched n lower panel n Fgure.4. It seems that ths vacuum nstablty problem appears n the SM, where the Hggs quartc self-couplng becomes negatve at energes above E SM 8 GeV to one-loop order n perturbaton theory. Ths so-called vacuum stablty problem s nvestgated at the end of the secton..5. Trvalty of QED Let us start to nvestgate the trvalty of QED. We wll nvestgate where the Landau pole of the QED couplng g s,.e. at whch energy scale the QED couplng blows up to nnty. The -functon of the QED couplng has been evaluated to fourth order of the couplng, 4, n Eq. (.) n Ref. [9], whch s = ; (.8) where () g() =(4) s the renormalzed ne structure constant whch s xed at. We can solve Eq. (.8 to lowest order n of the QED -functon. From ths we = 3 ) () () = ( ) ( ) d = 3 and therefore the QED runnng couplng s d ) () + ( ) = 3 ln ) (.83) 3 ln ; (.84) () = ( ) : (.85) 3 ( ) ln If we look at the couplng n Eq. (.85) then we can dentfy a pole at the momentum scale 3 3 L = exp ' 9 GeV exp ' 5 GeV; (.86) ( ) =8 Page 5 of 93

28 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS where we have used that (m Z = 9:9 GeV) =8 (n Ref. []). It looks lke that there s a pole at very large energy. Ths pole s the famous Feldman-Landau (F-L) ghost. We can not conclude that there s a Landau pole, because we can not use perturbaton theory anymore, when the ne structure constant s >. The physcal mechansm that works here s the phenomenon of charge screenng. There wll be created vrtual electron-postron par around the bare charge. The bare charge wll be surrounded of a cloud of vrtual charges whch wll reduces the value of the bare charge seen at large dstances. The bare charge wll be more and more vsble for hgher and hgher momentum appled. A dsaster wll therefore occurs when the couplng becomes nnte at a nte momentum scale. It looks lke that ths wll happen f ( ) s nonzero, then the pole leads to derent nconsstences n the QED theory. Someone can come to the concluson that QED s trval, because the theory s nconsstent except ( ) vanshes. Ths concluson s not warranted alone, because we have excluded the hgher order of the -functon n Eq. (.8). The -functon s also calculated perturbatvely as mentoned, and therefore t s lkely that the Landau pole s an artfact of the perturbaton theory. In the SM ths Landau pole s at L ' 34 GeV (accordng to Ref. [3]). In fact the F-L ghost pole does not appear before well beyond the Planck scale (E p = : 9 GeV), and therefore t seems that there s no problem..5. Trvalty of Hggs Sector Now, we wll nvestgate the trvalty n the Hggs sector wth the Hggs self-couplng whch s a scalar 4 eld theory. The -functon of ths eld theory s derved n Appendx E. The dagrams that contrbute to ths -functon of the Hggs self-couplng n only the Hggs sector to rst loop-order are shown n Fgure.5. Terms of the Lagrangan n Eq. (.6) that contrbutng to the dagrams are ( y ) = 4 h4 + ( )h + : : : ; (.87) where the would-be Goldstone bosons ;;3 also contrbute n the loop dagrams. The rst two dagrams are the rst loop correctons to the Hggs self-vertex, whle the last dagram s the rst loop correcton to the Hggs wave functon. The -functon s equal to the second term n Eq. (7.95) n Appendx E, whch s h h h = 4 (4) : (.88) h + + h h h h h Fgure.5: The dagrams that contrbute to the -functon of Hggs couplng to rst loop-order n Hggs sector wthout Yukawa couplngs. h Page 6 of 93

29 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS We can derve the runnng couplng () by solvng ths equaton. couplng s always postve to leadng order () d = d ) () ( ) ( ) () = ; ( ) log ( ) We obtan that the runnng = ln ln ) (.89) whch has a Landau pole. To determne ths Landau pole we need to know a value of the self-couplng at some energy scale. We have from Eq. (6) n Ref. [34] the MS top-yukawa couplng renormalzed at the top pole mass m t, whch we wll use n next subsecton, s gven by t (m t ) =: :557 mt :3 mh GeV 73:5 GeV s (m Z ) :84 :4 :th; :7 5 (.9) where there s a theoretcal error, :th, whch comes from non-perturbatve eects. The Hggs self-couplng n MS scheme whch s renormalzed at the pole top mass s also determned n Ref. [34] (Eq. (63)), whch s (m t ) =:577 + :5 mh GeV 5 :4 mt GeV 73:5 :4 th: (.9) ( = 7:44 GeV; g 3 = :7) Fgure.6: Left panel: The Landau pole n the runnng of the Hggs self-couplng, whch dverges at L = :8 5 GeV. Rght panel: The calculaton of the value of the strong couplng at the top mass g 3 (m t ) = :75 from the couplng renormalzed at the Z boson mass s (m Z ) = g 3=4 = :84 n Table.3. m h [GeV] m t [GeV] m Z [GeV] s (m Z ) 5:9 :4 Ref. [3] 7:44 :6 Ref. [75] 9:876 : Ref. [76] :84 :7 Ref. [33] Table.3: Physcal Constants wth uncertanty. Page 7 of 93

30 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS We can use ths expresson to determne the self-couplng to be (m t ) = :6 by nsertng the values n Table.3 nto Eq. (.9). Therefore, the Hggs self-couplng n Eq. (.89) wll ht a Landau pole at the energy scale L (m t ) log = ) 6 L = m t exp = 7:44 GeV exp (m t ) 4 :6 = 8:3 4 GeV: (.9) Ths Landau pole s shown n left panel n Fgure.6. We can not know wth certanty, that there s a Landau pole, because the perturbaton theory breaks down here. If the numercal calculatons seems to conrm that the Hggs quartc couplng dverges when the Yukawa couplngs vansh, then we can conclude that the couplng must be zero for the theory to be consstent. Thus, the Hggs sector s a trval theory, when the Hggs boson nteracts only wth tself..5.3 Vacuum Stablty n the SM Ths problem can be allevated by addng the Yukawa terms to the Hggs doublet term as n the SM, such that we have from Eq. (.6) and Eq. (.73) the Lagrangan terms ( y ) t j q Lj t R t j q L j t R : (.93) We have only ncluded the top-yukawa couplng, because the Yukawa couplng s proportonal to the fermon mass. Thus, the top-yukawa couplng contrbutes much more than the remanng Yukawa couplngs. The -functon for the Hggs self-couplng s derved n Appendx E, where both Hggs self- and top-yukawa nteracton are ncluded. The dagrams that contrbute to the -functon of the self-couplng s shown n Fgure.7. h h h h h h + t t + + t t h h h h h h h h h t t h + h h h h Fgure.7: The dagrams that contrbute to the -functon of the Hggs self-couplng n SM to rst loop-order. The rst two loop dagrams contrbute to the rst loop order correctons to the Hggs quartc self-vertex, whle the last two dagrams contrbute to the correctons to the Hggs wave functon. The -functon s Page 8 of 93

31 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS gven n Eq. (7.95) n Appendx E, = (4) (4 6 4 t + t ); (.94) where the rst term comes from the loop dagram and 4, the second term comes from loop dagram, whle the last term comes from loop dagram 3 n Fgure.7. If we couple the Hggs boson to the gauge bosons, then we obtan to rst loop-order (Eq. 3.5 n = (4) t + t 3g 9g g4 + (g + g ) ; (.95) where the -functon of the top-yukawa couplng to rst loop-order (Eq. 3.3 n Ref. [8]) s = 9 7 (4) 3 t g g + 8g 3 t ; (.96) and the gauge couplngs g, g and g 3 are assocated to the U()Y, SU()W and SU(3)C gauge symmetry, respectvely, whch have followng -functons to rst loop-order (Eq. 3. n Ref. = 4 96 g3 = 9 96 g3 ; = 7 6 g3 3: (.97) Fgure.8: The runnng of the Hggs self-couplng for derent top masses calculated by the Matlab. The yellow lne s the RG evoluton of for the top mass n Table.3. For the nner two lnes around the yellow lne the top mass s vared by, and the outer two lnes the top mass s vared by 5. Page 9 of 93

32 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS Accordng to the second term n Eq. (.95), f the quartc couplng s sucently small, then the top- Yukawa would domnate the -functon. Therefore, t can maybe drve the quartc couplng to negatve values such that the theory becomes unstable. If the quartc couplng s very large, then the trvalty problem above mght be relevant agan. To studyng the vacuum stablty we use Matlab to solve the coupled derental equatons n Eqs. (.95)-(.97) for the quartc couplng usng Euler's method. In Fgure.8 the runnng of the Hggs self-couplng are plotted for varous top masses calculated by Matlab. The top couplng t (m t ) and the Hggs self-couplng (m t ) renormalzed at the top mass are calculated by usng Eq. (.9) and Eq. (.9), respectvely, where the values n Table.3 are been used. The gauge couplngs g (m t ), g (m t ) and g 3 (m t ) renormalzed at the top mass are found by calculatng the RG evoluton of them as n rght panel n Fgure.6 for the strong couplng, g 3. For example, the the strong couplng at the top mass s found to be g 3 (m t ) = :75 from the couplng at the Z boson mass s (m Z ) = :84 n Table.3. The yellow lne n Fgure.8 s the RG evoluton of for the average value of the top mass n Table.3. For the nner two lnes around the yellow lne the top mass s vared by, and the outer two lnes the top mass s vared by 5. Ths plot shows that the vacuum of the SM to rst loop-order s unstable at energes above around 8 GeV. Accordng to Ref. [65], the nstablty scale s computed to be around 5 GeV for two-loop QCD and Yukawa correctons wth the central values n Table.3. Therefore, the nstablty scale s pushed up, when we nclude the next-to-next-to-leadng order (NNLO) loop correctons. Fgure.9: The runnng of the Hggs self-couplng for varous values of the strong ne constant and the Hggs mass calculated by Matlab. Left panel: For the nner two lnes and the outer two lnes the strong ne constant s vared by and 5 around the yellow lne wth the average value of s n Table.3, respectvely. Rght panel: The Hggs mass s vared by and 5 around the average value n Table.3, respectvely. In Fgure.9 the runnng of the self-couplng for varous values of s and the Hggs mass are plotted. In left panel the nner two lnes and the outer two lnes the strong ne constant s vared by and 5 around the yellow lne wth the average value of s n Table.3, respectvely. Whle n the rght panel the Hggs mass s vared by and 5 around the average value n Table.3. Thus, the RG Page 3 of 93

33 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS evoluton of s larger senstve to the varaton of the top mass than the Hggs mass and even smaller senstve to the strong ne constant. We have shown perturbatvely that although n solaton the SM Hggs sector s trval. Ths s moded when the top-yukawa couplng s ncluded. Instead, the SM s possbly vacuum unstable, because the Hggs self-couplng,, becomes negatve at energes above ts nstablty energy, E 8 GeV as computed n the one-loop approxmaton. In the upcomng sectons, we wll derve the rst loop-order correctons to the mass of the Hggs boson, where the quadratc dvergent correctons gve rse to a large ne-tunng problem of the Hggs mass. Ths becomes our motvaton to construct an underlyng model, whch tres to explan the ne-tunng of the Hggs mass by a dynamcal mechansm lke n QCD..6 Hggs Mass Correctons In the quantum vacuum, there s constantly produced partcle-antpartcle pars out of the vacuum, volatng the energy conservaton by takng the energy E from the vacuum for a short tme t, whch s possble accordng to Hesenberg's uncertanty prncple that says that Et < ~. These partcles are called vrtual partcles, and they are o-shell (E p 6= m ). When the Hggs boson propagates n the quantum vaccum, then t wll nteract wth these vrtual partcles. In the footnote, there s beng made a smple analogy to thermodynamcs as n Ref. [9], whch can help us understand the addtonal quantum contrbuton m h to the mass of the Hggs boson.3 The strength that the Hggs boson wll nteract wth any SM partcles s proportonal to the mass of the correspondng partcle. These nteractons result n correctons to the Hggs mass. The one-loop dagrams that contrbute to the Hggs mass are shown n Fgure.. We have the squared mass of the Hggs boson, m h, receves an addtonal contrbuton n the form: where m h s the physcal Hggs mass, m h m h = m h + m h =m h + k + ; (.98) s the bare Hggs mass, and s a cuto. We have only ncluded quadratc loop-contrbutons, because they contrbute mostly compared to the logarthmcally. We calculate later the coecent k to be :6 from the one-loop dagrams. We have that the physcal mass of the Hggs boson s very small compared to the Planck scale (the largest cut-o we can presently magne to the SM), and therefore the bare Hggs mass s needed to be ne-tuned extremely much. Thus, the Hggs partcle s specal, because there are no symmetres that protect aganst the quantum uctuatons. 3 We replace the quantum uctuaton of the vacuum wth the thermal uctuatons of a thermodynamc system wth a temperature, T. The partcles, P, n the thermodynamc system play the role of the vrtual partcles n the vacuum, and the temperature, T, corresponds to the cuto,. If we nsert another partcle, H, wthout momentum nto the thermal system, then we expect that the collsons of the partcles, P, wll soon brng the partcle, H, n thermal equlbrum. Therefore, the energy of the partcle, H, wll quckly become of order T. Ths s an analogy to what happens n the quantum system, here wll the Hggs mass (analogous to H) be pushed towards because of quantum uctuaton eects from the vrtual partcles. Page 3 of 93

34 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS h q p f f p q p q h h q p h; ' ; ' Z p q h h q W ; Z q h h p q Z ; ' Z; W p q q h h q W + W q h p q Fgure.: The one-loop dagrams that contrbute to the correcton of the Hggs mass. In followng we wll calculate the ampltudes of the loop-dagrams n Fgure.. Let us dene (q) as the sum of all one-partcle-rreducble nsertons nto the propagator,.e. that we have (q) = PI Then we have that the full two-pont functon for the Hggs propagator s gven by the geometrc seres d4 xhjth(x)h()je p x = PI = + + PI PI + : : : We can rewrte each Hggs propagator as =(q m h ) and express the above seres as d 4 xhjth(x)h()je q x = q m + h q m (q) h q m + h q m h X n (q) q m h q m = h q m (q): h n= (q) q m h + = (.99) Therefore, the Hggs mass s corrected by m h = m h m h = (q): (.) To calculatng ths mass correcton to rst loop-order, we need to calculate the sum of all one-partclerreducble dagrams for the Hggs propagator, whch are shown n Fgure.. To ths work, we wll Page 3 of 93

35 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS rstly calculate two useful ntegrals (calculated n Eq. (7.97) and Eq. (7.98) n Appendx C-). The rst ntegral s d4p () 4 p m f + = s + m f m f ln m f ln m f ; + + m f 3 A5 m f (.) where we have used Cauchy's resdue theorem to solve the ntegral, we have made a hard cuto at, and n the last step we assume that m f. The second useful ntegral s solved as follows d 4 p () 4 (p m f + )((p q) m f + ) 8 + where we have used followng dentons l p xq ) l = p + x q xqp; x( x)q + xm + ( x)m : dx ln ; (.) To solvng ths ntegral we have also carred out a Wck rotaton, where we make the substtutons l E = l and ~ l E = ~ l. Now, we are ready to calculate the one-loop dagrams n Fgure.. fermon loop = e s W 4m W We start wth the Hggs propagator wth a fermon loop whch s calculated n Eq. (7.99) n Appendx C-. The correcton gves m f d 4 p Tr((p= + m f )((p= q=) + m f ) () 4 (p m f + )((p (.3) q) m f + ) 4 e m f s W 4m W " + m 6 f ln + e m f s W x( # x)q dx ln + m f ; m f 4m W 4 (4m f m h ) where both the rst and the second ntegral n Eq. (.) and Eq. (.) are been used. The next dagram s the Hggs propagator wth a Hggs loop, whch gves the contrbuton h loop = e 3 m h d 4 p 4s W m W () 4 p m h + 3 e m h s W 4m W m 6 h ln ; (.4) m h where the rst ntegral n Eq. (.) s used. There are also the same knd of dagram wth both ' loops and a ' Z loop, whch gve ' loop = e m h d 4 p 4 m W () 4 p m W s W + e s W m h 4m W m 6 W ln ; (.5) m W and ' Z loop = e m h 4 s W m W d 4 p () 4 p m Z + e m h s W 4m W m 6 Z ln : (.6) m Z The next two dagrams are them wth a W and Z loop, whch gve Page 33 of 93

36 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS and Z loop = e c ws w W loop = e s w g d 4 p () 4 g d 4 p () 4 g p m Z g p m W 4 e s W 8 e s W m Z 4m W m W 4m W m 6 W ln ; (.7) m Z m 6 W ln : (.8) m W The dagrams wth a Z=' Z and a W =' (derved n Eq. (7.) and Eq. (7.) n Appendx C-) contrbute wth Z=' Z loop = e s W e s W d 4 p 4c w () ( p + q g q) (p q + q) 4 p m Z (p q) m Z m Z 6 + : : : ; 4m W (.9) and W =' loop = e s W e s W m W d 4 p 4m W () ( p + q g q) (p q + q) 4 p m W (p q) m W m W 6 + : : : : 4m W (.) The correcton terms whch are quadratc n are nterestng to consder, because they lead to a naturalness problem. We consder the dagram where the fermon n the fermon loop s a top quark, because t contrbutes much more than the other fermons because of ts large mass. The quadratc contrbutons are top =4 e s W m t 4m W 6 + : : : ; Hggs = h loop + ' loop + ' Z loop = = e s W 3m h 4m W 6 + : : : ; Z = Z loop + Z=' Z loop = [4 ] e s W m Z 4m W W = W loop + W =' loop = [8 ] e s W = e 3m W s W 4m W 6 + : : : : m W 4m W e m h s W 4m W 6 + : : : 6 + = e 3m Z s W 4m W 6 + : : : 6 + : : : ; (.) By nsertng the correctons n Eq. (.) nto Eq. (.), the squared mass of the Hggs boson, m h, receves an addtonal contrbuton m h = m h + m h =m h + e s W 3 4m 4m W 6 t m h m Z m W =m h + 3G F 8 p 4m t m h m Z m W m h + k + ; + + (.) Page 34 of 93

37 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS where the Ferm constant s p e G F : (.3) 8m W s W The result of the Hggs nteractons wth the vrtual partcles s that the squared mass of the Hggs boson m h receves an addtonal quantum contrbuton m h = k, where s the maxmum energy accessble to vrtual partcles and the constant k s the proportonalty constant n Eq. (.) whch s k 3G F = p 4 (4m t m W m Z m 3 :64 5 GeV h ) = p 8 h4(7:44 GeV) (8:39 GeV) (9:9 GeV) (5:9 GeV) (.4) =:57 :.7 The EW Herarchy Problem If we assume that the cut-o n Eq. (.) s the Planck mass = M P = : 9 GeV where we at least expect new physcs, then the so-called EW herarchy problem n the SM arses. The EW herarchy problem s that there s no scentc explanaton on why the weak force s very much stronger than gravty (the large gap between ther energy scales v EW =M Plack 7 ). We have namely that the physcal mass of the Hggs boson s very small compared to the Planck scale, and therefore the bare Hggs mass, m h =m h + km P = m h + :57 (: 9 GeV) = (5 GeV) ) m h =(5 GeV) ( 8 GeV) ; (.5) s needed to be ne-tuned extremely much. In the followng subsecton, we wll dene a possble quantty, whch s a measure of the ne-tunng of an observable compared to the parameters n a model..7. Fne-Tunng of Models In the followng, we wll brey gve a possble denton of ne-tunng n models. Fne-tunng refers to cases when the parameters of a model must be adjusted very precsely n order to agree wth observatons. By denng a quantty for the ne-tunng then we can compare the ne-tunng between the varous models. Ths quantty whch measures the amount of ne-tunng n any partcular parameter,, to produce the observable, O, s hstorcally been ntroduced by Barber and Gudce as the relatve rato between the observable and the parameters normalzed to them (cf. Eq. (36) n Ref. [45]),.e. O BG; Ths quantty gves a measure of ne-tunng for each < max; (.6) One possble way to nd the total ne-tunng n O could be by smply takng the maxmum of all the O BG,. We can decde to have the maxmum value max =. Ths means one percent change of the parameter,, gves rse to maxmal one hundred percents change of the observable O to have that the observable s not ne-tuned. Page 35 of 93

38 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS.7. Fne-Tunng of the Hggs Mass We can calculate the dened quantty for the ne-tunng n Eq. (.) for the mass of Hggs boson, m h, compared to ts tree-level mass, m h, where m h = m h k. For a cuto at Planck scale, = M P = : 9 GeV, ths quantty s m h BG;m h m h m h =:4 3 : m = h k = k = :57 (: 9 GeV) (5:9 GeV) m h m h (.7) Thus, the ne-tunng quantty s extremely large. Therefore, the Hggs mass, m h, s extremely ne-tuned compared to ts tree-level mass, m h. In the followng we follow Ref. [9], where t s tred to get an ntuton of how much the Hggs mass s ne-tuned by makng a smple analogy. It requres a steady hand to balance a pencl on ts tp on a table. If r s the radus of the tp surface and R s the length of the pencl, then the needed accuracy s of the order of r =R. We can compared ths accuracy to the ne-tunng quantty above. By usng that the radus of the tp s about r tp mm, whch gves that the length of the pencl s approxmatvely m h BG;m h R ) rtp R q m h BG;m h mm = p :4 3 mm = :5 3 m: (.8) The radus of the solar system s about R Solar System = 5 m. In Ref. [9] ths ne-tunng s compared to that we need to balance a pencl mnmum as long as the solar system on a tp of one mllmeter wde to reproduce the necessary accuracy G F =G N. 4 Ths makes that the SM seems unnatural, among others because ths enormous ne-tunng of the Hggs mass. The orgn of the Hggs mass n the SM s complete unclear. Ths ndcates a need for a more general mechansm or some symmetres that provde a ratonalzaton for the Hggs boson. Ths leads us to beleve that there could be a theory beyond the SM. It s a puzzle why the Hggs boson should be lght, when the nteractons between t and SM partcles would tend to make t very heavy. It can e.g be cured f the Hggs s a composte partcle of some new dynamcs such that the cuto s low wth respect to the weak scale and there s consequently only a small amount of ne tunng. Ths composte dynamcs can be qute smlar to QCD, where color and quarks wll be conned whch wll be observed as hadrons (mesons and baryons). Ths wll happen below a typcal scale, the cuto, lke QCD energy scale QCD. Another way to cure the ne-tunng problem s a supersymmetrc model (e.g. n Ref. [?]). Supersymmetry would lnk the fermons and the bosons by a transformaton that take a fermon or a boson nto a boson or a fermon. Therefore, the supersymmetry predcts a superpartner partcle for each partcle n the SM. Physcal the loop contrbutons to the Hggs mass sum to zero for energes above the supersymmetrc breakng scale. The superpartner partcles predcted would cancel out the contrbutons to the Hggs mass from ther SM partners, because ther 4 We have that the Hggs mass s mh G = F and the Planck mass s MP = G = N, where GF and GN s Ferm constant and the gravtatonal constant, respectvely. Page 36 of 93

39 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS contrbutons have same sze and opposte sgn (Ref. [66]). Thus, the proportonalty constant k n Eq. (.) would become zero above the the supersymmetrc breakng scale, whch would remove the ne-tunng n Eq. (.7). In ths thess, we wll focus on models of the Hggs mechansm, where the Hggs boson s dynamcally produced as ether a composte or a partally composte partcle consstng of new fermons, called technquarks. The possble dynamcs that forms the Hggs boson can be smlar to QCD. Thus, n the next secton we wll examne the mechansm of the chral symmetry breakng n QCD, where the quarks conne to the hadrons..8 Chral Symmetry Breakng n QCD We wll examne the mechansm of chral symmetry breakng n QCD, whch dynamcally generates the masses of the hadrons. We wll construct an eectve theory for the lghtest pseudoscalar-mesons consstng of the three lghtest quarks n the SM and dentfy ther masses n the eectve Lagrangan. We wll start by consderng phenomena n the theory of the strong nteracton of elementary partcles, Quantum Chromodynamcs (QCD)..8. Quantum Chromodynamcs (QCD) The theory of the strong nteracton of elementary partcles, called QCD, s a non-abelan gauge theory wth SU(3)C as gauge group. The correspondng charges to ths SU(3)C are called colors. The quarks have besdes the avor also color and transform as the fundamental representaton of color SU (3)C. The eght colored gauge bosons, called gluons, are n the adjont representaton of SU(3)C. The dentons of a fundamental and adjont representaton of a symmetry group are explaned n Appendx C. We wll start wrtng down the Lagrangan of QCD, whch s wrtten as L QCD = A =D j j A 4 Ga G a = A =D j j A Tr[G G ]; (.9) where j A s the quark spnor wth color ndex j = ; ; 3 and avor ndex A, and the covarant dervatve can be wrtten as D j j g s G a T a j; (.) where Ta j = a j = are the generators of SU(3) C,.e. a j are the Gell-Mann matrces wth a = ; : : : ; 8. The gluon eld strength tensor s G where f j ab c =@ G G j g s [G ; G ] j G d =@ G c T j T G c j c g s G a G b f c T j G d ab Tc j = G c T j are the structure constants. Therefore, we have T j d g s G a G b [T a; T b ] j c ; (.) G c G G c gg a G b fab c : (.) Page 37 of 93

40 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS We have wrtten the gluon elds as G j = Ga Ta j, wheren the two ndces ; j s shfted wth respect to each other, because s an ant-fundamental ndex and j s a fundamental ndex. g g g f f f f f Fgure.: The one-loop correcton dagrams to the QCD vertex. To rst loop-order the -functon of the QCD couplng g s (cf. (3..) n Ref. [3]) s (g gs 3 s ) = (4) + O(g5 s ) = N gs 3 3 f (4) + O(g5 s ); (.3) whch s calculated from e.g. the two one-loop dagrams n Fgure.. We can derve the runnng couplng by solvng the followng derental equaton dg s d ln = g 3 s (4) ) g s = g s + g s 8 log Q ; (.4) where g s = g s () and g s = g s (Q). If we have that s () = g s=4 and s (Q) = g s=4, then we obtan s (Q) = s () : + Q (.5) s() log Connement s Hadrons Asymptotc freedom QCD MeV Fgure.: QCD runnng couplng as functon of the energy. Page 38 of 93

41 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS In Fgure. the runnng couplng s (Q) s plotted as functon of energy Q. At hgh energy or hgh momentum transfers we have that the QCD couplng s small. We say that QCD s asymptotcally free. We can see that QCD s asymptotcally free f the number of quarks s not too bg ( > ), when the number of quarks s N f < 33=, whch s met wth N f = 6 quarks n the SM. Therefore, QCD s treatable by perturbatve methods at hgh energy. We can also observed that there s a specc energy scale, QCD, where the couplng blows up. Therefore, QCD becomes strongly nteractng at low energy. The condton for ths phenomena called QCD connement s = s () log QCD : (.6) By nsertng ths condton nto Eq. (.5) we obtan ths form of the QCD couplng s (Q) = ; Q log QCD (.7) where = 3 N f = 7 n the SM, because there are sx quarks n the SM. Thus, we have that the runnng constant of QCD grows for decreasng energy. By dmensonal transmutaton the nteracton may be charactersed by the dmensonful parameter, QCD, namely the value of the RG scale at whch the couplng constant dverges. Dmensonal transmutaton s a physcal mechansm provdng a lnkage between a dmensonless parameter (e.g. the QCD couplng g s ) and a dmensonful parameter (the energy scale QCD). Below ths QCD scale, a connement of quarks and gluons n hadrons happens below around GeV. Perturbaton theory, whch produced the runnng formula above, s only vald for a couplng s. Accordng to lattce calculatons and experments, there exsts such a QCD scale, whch s QCD = MeV (n Ref. [37]), whch s an nfrared cuto (Q QCD mples s ). The masses of the hadron resonances are n the order of ths scale. The energy scale,, n Eq. (.) can maybe have a natural orgn relatve to the Planck scale, f t s explaned by a dynamcal mechansm as the QCD scale above. On the other hand, n the case of theores such as QED, s an ultravolet cuto (Q mples ) at whch the Landau pole happens as shown n left panel n Fgure.6. However, t seems that the Landau pole takes place long above Planck scale, therefore there are no problems. The hadron dynamcs at low energes can be nvestgated by performng nonperturbatve numercal computatons, e.g. by lattce QCD or by the method of eectve eld theores. The eectve eld theores, e.g. the non-lnear model, s based on at low energes a descrpton of the strong nteracton drectly n terms of the lght hadrons, e.g. the pseudoscalar mesons, s possble. Now, we wll descrbes how to construct such eectve eld theores..8. Constructon of an Eectve Lagrangan In ths subsecton, we wll talk about how we can construct an eectve Lagrangan, whch can descrbe the composte partcles consstng of quarks n QCD or the Hggs boson as composte partcle consstng of technquarks. When studyng a physcal system t s often the case that there s not enough nformaton Page 39 of 93

42 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS about a fundamental descrpton of some of ts propertes, e.g. when the perturbaton theory breaks down at low energes n QCD. In these cases we need to parameterze the correspondng eects by wrtng new nteractons wth coecents whch can be determned phenomenologcally. Expermental measurements of these parameters can hopefully provde the nformaton needed to provde a better descrpton of the propertes of the model. For dong ths, we need to determne the dynamcally degrees of freedom nvolved, and the symmetres they obeyed. Thereafter, we construct the eectve Lagrangan for these degrees of freedom whch wll respect the requred symmetres. It s mportant to have n mnd that the relevant degrees of freedom can change wth energy scale (e.g. mesons are a good descrpton of low-energy QCD, but not at hgher energes where we need to use quarks and gluons), and the physcs can respect derent symmetres at derent energy scales. Thus, the eectve Lagrangan s applcable only for a lmted range of energy scales. It s often that there s an energy scale, where for energes above the eectve Lagrangan does not work anymore. Ths method of eectve theores s straghtforward, and most mportantly t works. To begn wth, we can concentrate us about the Lagrangan of QCD (strong nteractons), whch (cf. Eq. (.9)) can be wrtten L QCD = A where G a s the gluon eld strength tensor and A =D j j A 4 Ga G a ; (.8) are the quark elds. We can wrte such a compact form of QCD because of gauge nvarance and renormalzaton. As menton, the quarks comnes hadrons (mesons and baryons) below an energy about one GeV, whch s set by the QCD cuto, QCD MeV. At ths energy scale, t s not possble yet to calculate QCD results exact, because the QCD couplng constant has been too large that we can use perturbaton theory anymore. Therefore, we want to make an eectve theory for these composte partcles. In ths case we are nterested n the descrpton of the nteractons among the lghtest composte partcles (e.g. the mesons n QCD). The most convenent parameterzaton of these degrees of freedom s n terms of the nonlnear untary eld (exponental parameterzaton) such that a (x)x a U (x) = exp ; (.9) f where a (x) denote the Goldstone elds (the lghtest meson elds n QCD) comng from the global spontaneously breakng pattern G! H, X a are the generators of the broken global symmetres, f s a constant (related to the pon decay constant n QCD), whch has unt of energy and therefore makes the argument of exponental untless. We have also that UU y = n, because the generators are hermtan. The quantty U transform under G global group as U! gug y, where g G. The eectve Lagrangan must obey ths global symmetry G of the fundamental Lagrangan, gauge symmetry nvarance, Lorentz nvarance, C nvarance and P nvarance. The eectve Lagrangan wth a cuto as n QCD, there s no actual ultravolet dvergences n most eectve Lagrangan computatons. Therefore, we have not necessarly the renormalzaton as gude lne to constructng the eectve Lagrangan terms. Page 4 of 93

43 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS Wth these constrants the eectve Lagrangan takes the form L e =Tr[@ U U ] + Tr[U@ U y U@ U y ]+ Tr[@ U@ U U@ U y ] + Innte Many Terms; (.3) whch s nvarant under the global symmetry, Lorentz, C and P transformatons. There can not be dervatve-free terms, because Tr[UU y ] = Tr[ n ] = n s a constant. We have the s p n momentum space, and t follows from dmensonal analyss that the coecent of an operator wth k dervatves behaves as = k 4, whre s a mass scale whch depends on the specc theory. Thus, the eect of an n-dervatve vertex s of order p k = k 4, and thus at an energy small compared to, then large-k terms have a very small eect. Therefore, the nnte many terms would become less and less mportant, and then we can make a perturbatve expanson n dervatve at sucently low energes. In next subsecton, we wll construct such a low-energy eectve theory for the eght pseudoscalar mesons, a, n QCD consstng of the three lghtest quarks n the SM. Problems happen when we have theores wth heaver partcles than the Goldstone bosons, other scalar exctatons. In ths case, we can arrange the elds as follows a (x)t a U (x) = S(x) exp f (.3) wth the heaver elds S(x), whch gves us that UU y = S y S 6= n. Ths gves terms whch are dervatvefree, because they are not constant n ths case. Therefore, we have the extra terms wth the form X n Tr[UU y ] n : (.3) Ths s a sum of nnte many terms, where each term s equally mportant. Therefore, t breaks down when we add the scalars together wth the Goldstone bosons. For example, we can construct a model lke QCD wth a composte Hggs consstng of smaller consttuents, where Hggs s a scalar partcle. For example, we can substtute the M matrx nto the U place. In ths case, such a model wll produce vertces wth many Hggs eld, h, external lnes. However, we can gnore many-h-vertces: Frstly t s hard to produce, and secondly at a gven number of h n the vertex then the energy would be above the energy scale where h would fall apart. Now, we wll construct such an eectve eld theory n QCD for the lghtest pseudoscalar mesons whch consst of the three lghtest quarks (the up, the down and the strange quark)..8.3 Chral Symmetry Breakng The connecton between the fundamental QCD Lagrangan and the low-energy eectve theores s constructed by symmetres of the sector of the lghtest quarks (e.g. the three lghtest u,d and s quarks), whch appear when masses of these quarks vansh (the chral lmt). The mass term n the QCD Lagrangan s L QCD,m = X q m q q (x) q (x); (.33) Page 4 of 93

44 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS where m q s the masses of the quarks q = u; d; s;. If the N f quark masses are equal, then the Lagrangan s nvarant under a SU(N f )V symmetry! exp( a a =), where a are the SU (N f ) generators. Ths symmetry leads to the conserved vector currents j a (x) = (x) a (x): (.34) If the quark masses vansh we have also that the axal SU(N f )A transformatons! exp( a 5 a =) s a symmetry. Ths symmetry leads to the conserved axal-vector currents j a 5; (x) = (x) 5 a (x): (.35) We have also the snglet vector (U()V symmetry) and axal-vector currents (U()A symmetry) j (x) = (x) (x); (.36) j 5; (x) = (x) 5 (x); (.37) where s the unt vector n quark avor space. These global symmetres are stll symmetres, even though that the quark masses are derent. The charges of these currents (n Eqs. (.34)-(.37)) generate the global group SU(N f )V SU(N f )A U()V U()A = SU(N f )L SU(N f )R U()L U()R; (.38) where t s very convenent to consder besdes vector and axal currents also the chral currents, j L; (x) = [j (x) j 5; (x)]; j R; (x) = [j (x) + j 5; (x)]; (.39) jl; (x) = [j (x) j 5; (x)]; j R; (x) = [j (x) + j 5; (x)]; (.4) whch have the symmetres SU(N f )L, SU(N f )R, U()L and U()R, respectvely. The global avor symmetry of the QCD Lagrangan n Eq. (.9) s U(N f )L U(N f )R =SU(N f )L SU(N f )R U()L U()R (.4) =SU(N f )V SU(N f )A U()V U()A: (.4) where SU(N f )V gves conservaton of the strong sospn, and U()V gves the conservaton of the baryon number. When we quantze the theory then the global group U()A s broken, because t s not anomalyfree (.e. the measures n the Feynman path ntegrals are not nvarant under these transformatons). The rest of the chral symmetry wth dmenson N f s spontaneously broken to a subgroup of only the vector symmetres wth dmenson N f as follows SU(N f )V SU(N f )A U()V! SU(N f )V U()V: (.43) Ths global symmetry of the Lagrangan wth the quark mass terms s an approxmatate symmetry, because the quarks have masses. For the lghtest three quarks, we can assume that they are approxmately Page 4 of 93

45 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS massless, because ther masses are somewhat less than the QCD scale QCD. For the up, down and strange quark we have m u m d ; QCD QCD and m s QCD < : (.44) If the quark masses are derent then the global symmetry s only U() U(), because the sospn symmetry SU(N f )V s broken to U(). Because of Eq. (.44) we have that the QCD Lagrangan has the approxmate global symmetry n Eq. (.43) for the three lghtest quarks (.e. N f = 3). The Vafa-Wtten theorem shows that vector global symmetres such as strong sospn (SU (N f )V charge) and baryon number (U()V charge) n vectoral gauge theores lke QCD cannot be spontaneously broken (cf. Ref. [46]). Therefore, the vectoral symmetres are unbroken after the spontaneously breakng n Eq. (.43). Accordng to ths theorem, the spontaneous breakng can maxmally break SU (N f )A, whch generates N f pseudoscalar Goldstone bosons. E.g. n the case where there are three massless quarks, then we can dentfy eght pseudoscalar Goldstone bosons, +,,, K +, K, K, K and 8. The explct breakng of chral symmetry by the QCD quark mass terms wll generates ther expermentally observed masses. The ampltude of the producton of a pseudoscalar meson state jp; b from the vacuum j (cf. page 59 n Ref. [3]) s hjj a 5jp; b = p f a P S ab ; (.45) where a; b = ; : : : ; N f. Ths transton ampltude contans the order parameter, the pseudoscalarmeson decay constant fp a S, whch s non-zero f the global symmetry s broken. The eectve Lagrangan can be expanded n a sere n the number of the pseudoscalar mesons elds ( a elds) as n Eq. (.3) as follows L e = L () e + L(4) e + : (.46) whch corresonds to an expanson n the momentum n momentum space. The rst term s L () e a : (.47) There can not be constructed a 3 term whch s Lorentz- and avor-nvarant. Therefore the rst nteracton term s a 4 term. In the followng, we construct an eectve Lagrangan for the eght pseudoscalar mesons consstng of the three lghtest quarks. The eectve Lagrangan can be expressed n terms of the exponentals of the elds as n Eq. (.9). Ths model s called the non-lnear model. The elds are wrtten as (x) U (x) = exp ; (.48) where we have the eld f PS Page 43 of 93

46 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS (x) = a (x) a = p p 3 + p K + p p 3 + p 8 3 K p p K K 8 p 3 C A : (.49) The broken generators, a, are the eght Gell-Mann matrces. The Lagrangan of the non-lnear model s constructed from U (x) n Eq. (.48) such that t reproduces the term n Eq. (.47). Ths ts wth whch s nvarant under the transformaton L = f P S 4 Tr[(@ U )(@ U y )]; (.5) U (x) = g L U (x)g y R ; (.5) where g L and g R are the elements of SU(N f )L and SU(N f )R groups, respectvely. We can expand the L further to hgher-order terms. The quartc term s X = L P =L Tr[(@ U )(@ U ) y ] + L Tr[(@ U )(@ U ) y ]Tr[(@ U )(@ U ) y ]+ L (4) L 3 Tr[(@ U )(@ U ) y (@ U )(@ U ) y ] + ; (.5) where the L are free parameters. In realty the masses of the lght quarks do not vansh. The chral symmetry s broken explctly by the QCD mass term n Eq. (.33). We nduces a correspondng term n the non-lnear model. A sutable ansatz can be L ;SB = f P S 4 B Tr[M q (U + U y )]; (.53) where B s a free constant and M q s the mass matrx of the lght quarks. By expandng the exponental of the pseudoscalar elds U (x) (Eq. (.48)) n the symmetry breakng term (Eq. (.53)) to second order and leavng out the constant term gves the mass term of the pseudoscalar mesons, L f P S ;SB = B Tr[M q (U + U y f P S )] = B Tr + a (x) a b (x) b 4 4 fp + O( 4) S L ;M = B m 3 u K + K p3 + m 3 d K K ) 3 8 p3 + m s K + K + K K : (.54) After dagonalzaton of the mxng terms of the elds 3 and 8 n Eq. (7.) n Appendx C-, we obtan the masses M =(m u + m d )B ; M K = (m u + m s )B ; M K = (m d + m s )B ; M = m u + m d M = mu + m d + 4m s 3 (m u m d ) (m s m u m d ) + (m u m d ) (m s m u m d ) B + O (m u m d ) 3 ; (.55) B + O (m u m d ) 3 : Page 44 of 93

47 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS The expectaton value of the mass term n Eq. (.53) s D f P S E B Tr[M q (U + U y f P S )] = B Tr[M q 3 ] = fp SB (m u + m d + m s ); (.56) and accordng to Eq. (.33) we have that the expectaton value of the mass term at fundamental level s D X By settng these mass terms equal each other, we obtan q q m q q E = m u h u u m d h d d m s h s s : (.57) m u h u u + m d h d d + m s h s s = fp SB (m u + m d + m s ) ) (.58) hj q q j = fp SB no sum over q: We have the quark condensate s related to the squared of the GB masses M GB n the Gell-Mann-Oakes- Renner (GMOR) relaton (cf. Eq. () n Ref. [47]), whch s wrtten as follows M GBf P S = X q h q q m q : (.59) By nsertng Eq. (.58) nto GMOR relaton above we obtan that the masses of the pseudoscalar mesons are X MGB = m q B ; (.6) whch are consstent wth the lghtest GB masses n Eq. (.55). In the next secton, we wll transfer the way to produce the masses of Goldstone bosons to a smple techncolor model. It could be a QCD-lke theory wth typcal energy scales n the order of TeV wth bound states of new knd of fermons whch provde the SM Hggs boson. In these models, the SM Hggs boson acheves ts mass from a dynamcal mechansm lke n QCD, where the masses are only aected logarthmcally by quantum correctons. Thus, the EW herarchy problem would not exst. q.9 Techncolor Models The dea of techncolor s that the EW herarchy problem assocated wth the mass of the Hggs boson can be evaded f the Hggs boson s not an elementary partcle but a composte object. If ths object are made of consttuents whch have masses only aected logarthmcally by quantum correctons, then the EW herarchy problem would not exst. Such models requre that the nteractons are strong such that the Hggs can be a bound state, and therefore we need to apply non-perturbaton theory. Thus, the Hggs boson would appear as an elementary partcle only at energes sgncantly below ts bndng energy. Ths constructon s actually rather ntutve, because a smlar constructon s realzed n the SM already. In QCD, there are bound states of quarks as consdered n prevous secton whch have the same quantum numbers as the Hggs and can nduce the breakng of the EW symmetry. The QCD condensates Page 45 of 93

48 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS wth the quantum numbers of the Hggs condensate can be constructed, but the challenge s that the QCD scale ( QCD MeV) s not sucent to provde the observed breakng of the EW symmetry (v EW = 46 GeV). Therefore, the smplest extenson to the SM could be a QCD-lke theory wth typcal energy scales n the order of TeV wth bound states whch provde the SM Hggs boson. These theores are called techncolor theores..9. Smple Techncolor The smplest verson of techncolor model s a QCD-lke theory at hgher energy scale. In ths smple techncolor model, we have a gauge group SU(N TC ) wth N f addtonal fermons Q, called technquarks. These technquarks are placed n the fundamental representaton of the gauge group SU (N TC ), whch are massless at tree-level. Thus, there are N TC techncolors. In addton, there are the N TC gauge bosons, called techngluons. Therefore, the total gauge group whch s an extenson of the standard model s SU(N TC )TC SU(3)C SU()W U()Y. Such a theory looks very much lke QCD, except t may have possbly a derent number of colors and avors. Its dynamcs wll be qute smlar to QCD, where techngluons and technquarks wll be conned, and technquarks can only be observed bound n technhadrons. Ths dynamcs wll be determned by a typcal scale, e.g. n QCD ths scale s QCD MeV. In the techncolor model there exsts also such a scale TC. Ths scale must be of the same sze as the electroweak scale, otherwse the EW herarchy problem wll emerge agan. We assume that ths scale s of the sze of TeV nstead of GeV lke n QCD. The technquarks are so far approxmately massless smlar wth QCD. Because they are massless then there s a chral symmetry that generates the global group SU(N f )V SU(N f )A U()V U()A. As n QCD, because of the dynamcs of the techngluons the chral symmetry of the technquarks s spontaneously broken to the global group SU(N f )V U()V. Ths gves us N f goldstone bosons smlar to pons and the other pseudoscalar mesons n QCD whch are pseudoscalar bound states of a technquark Q and an ant-technquark Q. The condensate wll have a sze of about TC as we can observe n QCD, whch gve the technquark an eectve dynamcal mass of the order of TC. If the technquarks have quantum number as the ordnary quarks, the technpons wll just have the same quantum numbers such that they can become the longtudnal degrees of freedom of the weak gauge bosons nstead of the would-be bosons n the Hggs mechansm. The Hggs s not one of the Goldstone bosons, but t wll be a scalar meson whch s the analogue of the meson of QCD. It s expected to be more massve and also more unstable than the Goldstone mesons. Snce a techncolor theory s based upon an analogy wth the dynamcs of QCD, then we can rescale QCD to determne the propertes of the pure techncolor theory. The man scalng rules (Eq. (.3) n Ref. [5]) are f QCD P S p N C QCD; hq Q j QCD j N C 3 QCD; m QCD; (.6) Page 46 of 93

49 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS where hq Q j QCD s the QCD condensate, f QCD P S 93 MeV s the pseudoscalar-meson decay constant n QCD n Eq. (.45), N C s the number of colors, QCD MeV s the QCD energy scale, and m s dynamcally generated consttuent mass. From the rst scalng rule above we can obtan a relaton for the technpon decay constant, whch s f TC P S r NTC 3 TC QCD The TC eectve gauge-knetc Lagrangan as n Eq. (.5) takes the form Q L =DQ L + Q R =DQ R! f P TC S 4 f QCD P S : (.6) Tr[D U (D U ) y ]: (.63) where Q L;R are the left-handed and rght-handed the technquarks, respectvely. Therefore, f N D doublets carry weak charges, then there are N D terms of the form above. Thus, the weak scale becomes v EW = p ND fp TC S. By nsertng the TC decay constant n Eq. (.6) nto ths expresson for the weak scale, we obtan v EW r ND N TC 3 TC QCD f QCD P S : (.64) From ths expresson we can solate the TC energy scale and determne the TC decay constant r r 3 v TC EW 3 N D N TC f QCD QCD :7 TeV; N D N TC where v EW = 46 GeV, QCD 5 MeV and f QCD P S P S we can determne the key propertes of the man classes of TC theores. f TC P S v EW r N D ; (.65) = 9:3 MeV (n Ref. [38]). Wth these scalng rules, As we wll see, the Goldstone bosons may not be used as the Hggs boson n a techncolor model. Therefore, we need to dentfy the Hggs boson as the scalar n QCD. In the followng, t gves rse to a problem, because we can not produce a techn- wth the Hggs mass m h = 5 GeV. We have that the techn-pon decay constant (f TC ) s the same as the EW energy scale (v EW ) for one doublet. We can determne f TC n Eq. (.6), where we expect f TC by scalng the pon decay constant constant n QCD up to EW scale wth the scalng rule f TC / p N TC and f QCD = v EW = 46 GeV = f QCD ; (.66) / p N C accordng to the scale rule n Eq. (.6), where N TC and N C = 3 are the number of colors n the TC model and colors n QCD, respectvely. By knowng that f QCD PS = 9:3 MeV (cf. Ref. [38]), we have f the techncolor guage group s SU(N TC ). = 665 r NC N TC ; (.67) Thus, the Hggs boson can be compared to the lghtest composte scalar state f (5) or also called n QCD, such that the Hggs s a techn-. In ths case the mass of the Hggs boson scales as follows (as n Ref. []) r 3 m h = m = (7 to 47) GeV; (.68) N TC Page 47 of 93

50 CHAPTER. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS where the mass of the scalar meson f (5) or s m = (4 55) GeV (cf. Partcle Data Group). By rescalng ths resonance n QCD to techncolor n Eq. (.68), t s too heavy to be the observed Hggs boson wth 5 GeV mass, unless the number of techncolor s very hgh. Ths n turn s constraned by electroweak precson measurements as follows n the next chapter. Ths s one of the dcultes wth the Hggs mechansm as a techncolor model. In the followng chapters, we wll experence that there are other problems wth the techncolor models than ths one, e.g. the generaton of the SM fermon masses n the composte dynamcs, avor-changng neutral currents and the constructon of the mass herarchy between the fermons.. Chapter Concluson We can conclude that the Hggs boson s responsble for the orgn mass of the elementary partcles n the SM and for curng the would-be volaton of untarty n the weak sector. We can also conclude, that the custodal symmetry s mnmal broken by the Yukawa sector n the SM. Accordng to expermental data from LEP experments, the parameters descrbng the untarty and the breakng of the custodal symmetry are W = :9 +: : and = :6 :9, respectvely. These measurements are consstent wth the SM, where both parameters are normalzed to be one. Frstly, we can conclude, there are stll room to a new Hggs doublet or other physcs beyond the SM whch s responsble for untarty of the scatterng ampltudes. Secondly, we have that new physcs beyond the SM should be custotal symmmetrc or the symmetry must be broken mnmal. Furthermore, we can conclude that the SM model seems to be a non-trval theory wthout Landau poles and may to be unstable wth an nstablty energy at E 8 GeV to one-loop perturbatve calculatons. The SM may not be a complete theory of the EW symmetry breakng accordng to the calculatons of the Hggs mass correctons, because the Hggs mass s very ne-tuned. Our man motvaton s to search after a possble dynamcal orgn of the EW scale whch would be natural. Thus, we tred smply to construct a rescaled QCD model. In these models the Hggs boson s a composte resonance consstng of technquarks lke the hadrons n QCD. These knd of models have a natural cuto scale, whch s explaned by an underlyng dynamcal mechansm lke the energy scale QCD n QCD. One of the dcultes to establsh these techncolor models s that t s heard to explan the mass of the observed 5 GeV Hggs boson, unless the number of techncolor s very hgh. As follows n the next chapter, the electroweak precson measurements at the LEP experments constran the possble number of techncolor. Page 48 of 93

51 Chapter 3 Mnmal Walkng Techncolor The Mnmal Walkng Techncolor (MWT) theory s proposed n Ref. [4] and eectve Lagrangan s developed n Ref. [5], where we would mostly follow Ref. [] n ths chapter. In ths theory, we have the extended gauge group SU()TC SU(3)C SU()W U()Y. The elds of the techncolor SU()TC gauge group are the technfermons U L, D L, U R and D R, and techngluons whch all transform accordng to the adjont representaton of SU()TC as descrbed n Appendx C. The composte secton of the model suers from the Wtten topologcal anomaly, because there are an odd number of left-handed fermon doublets under the weak gauge group, snce there are three Q L (U L ; D L ) doublets. The Wtten topologcal anomaly s explaned n Appendx H. The model s cured by addng a new fermonc weak doublet L L, whch are snglets under techncolor gauge group. Furthermore, the weak snglets N R and E R are ntroduced to cancel the gauge anomales wth the hypercharge assgnment n Table 3 as shown n Appendx B, where the parameter y can take any real value. We refer to the elds as L L, N R and E R the New Leptons. Q L = L L = Feld SU()TC SU(3)C SU()W U()Y U L D L C A 3 y U R 3 y+ D R 3 y N L E L C A 3 y N R 3y+ E R 3y Table 3.: Representatons of fermons n MWT under SU()TC, SU(3)C, SU()W and U()Y. In the analyss of the eectve Lagrangan wth the global symmetry SU(4), we assume that ths SU(4) 49

52 CHAPTER 3. MINIMAL WALKING TECHNICOLOR symmetry spontaneously breaks to SO(4), because the condensate hu R U L + D R D L s only nvarant under SO(4) SU(4). In ths model the EW symmetry breaks smultaneously wth the chral symmetry, because we can nd SU()W U()Y SU(4) and U()Q SO(4). In ths chral symmetry breakng there s also found a trplet of GBs whch are absorbed as the longtudnal components of the weak gauge bosons lke n the Hggs model. Addtonally, there are sx Goldstone bosons. The lghtest scalar exctaton around the vev can be dented wth a possble Hggs canddate as n the smple TC model n prevous chapter. Fnally, we can dentfy a custodal symmetry n the unbroken symmetry group SU () C SO(4). After we have constructed the eectve Lagrangan and derved the spectrum of the masses of the composte scalars and vectors n the theory, we dscuss how the fermons obtan ther masses from an extended techncolor (ETC) theory wth a hgher symmetry group SU (N ETC ). Subsequently, t s shown that a walkng theory can allevate the potental problems whch such an ETC theory creates. In such a theory the couplng of the gauge theory s nearly constant from the scale TC to ETC, whch s determned by the number of colors and fermon avors f the theory s a QED-lke, QCD-lke, Banks- Zaks or Walkng theory. The walkng dynamcs was rst ntroduced n Refs. [5]. Therefore, the theory s called Mnmal Walkng Techncolor, mnmal because we have the mnmal number of technfermons gauged under the EW group (only two technfermons), walkng because the couplng s constant n a wde range of energy, and techncolor because the vacuum of the theory s algned n the techncolor lmt where the EW symmetry breakng happens at the same scale as the chral symmetry breakng. In prncple, t s possble to have more fermons whch have no EW nteractons, and such that t does not contrbute to the EW precson parameter, the S parameter, cf. Ref. [6]. At the end of the chapter, we provde the lnk between the theory at the underlyng and eectve Lagrangan level va the Wenberg sum rules (WSRs) n the case of runnng or walkng dynamcs by followng Ref. [6, 4, 4]. Runnng dynamcs means here that the couplng has a dependence of energy whch s smlar the one n QCD or MWT,.e. asymptotcally free gauge theores. In the walkng dynamcs the couplng s nearly constant for a wde range of energy thereafter the runnng behavour resumes at hgh energes. In same secton, we derve a formula for the EW parameter called S, whch can be calculated from the underlyng Lagrangan and thereafter be compared wth the expermental lmts of the S parameter. 3. The Underlyng Lagrangan for Mnmal Walkng Techncolor We consder a new dynamcal sector whch s an SU()TC techncolor gauge theory wth two technfermons. The two technfermons, whch are n the adjont representaton of the SU ()TC techncolor gauge group, can be wrtten as U U L;;a U _;a R A and D D L;;a D _;a R A ; (3.) Page 5 of 93

53 CHAPTER 3. MINIMAL WALKING TECHNICOLOR where ; _ = ; are Lorentz ndces, and a s the adjont gauge ndex of the gauge group of the theory. We have the Weyl spnors U L;;a and D L;;a whch are the left-handed technup and techndown, and U _;a R and D _;a R whch are the correspondng rght-handed partcles wth techncolor ndex a. Accordng to Fgure 3.5 explaned later, we must have that < n Eq. (3.9) to get a theory whch s asymptotcally free. We have that = 4 3 T R 3 C A < ) N f < =4 = :75; (3.) where C A = and T R = N f = when the technfermons and techngluons are n the adjont representaton of SU()TC. Thus, the theory s asymptotcally free f the number of avors s less than :75, whch s the case n ths theory. The two left-handed adjont technquarks and the two rght-handed technquarks can be wrtten n a doublet and two snglets of the EW group as Drac spnors nstead of Weyl spnors n Eq. (3.), respectvely, Q a L U a L D a L where a s the adjont color ndex of the gauge group SU()TC. A ; U a R ; DR; a a = ; ; 3; (3.3) These left-handed technquarks are arranged n three weakly charged doublets. So far, the model suers from the Wtten anomaly accordng to Appendx H. We have that an SU()TC gauge theory s mathematcally nconsstent f there are an odd number of left-handed doublets and no other representatons n ths theory. However, ths can be solved by addng a new weakly charged leptonc doublet and ther rght-handed snglets, whch can be wrtten as L L N L E L A ; NR ; E R ; (3.4) whch are techncolor snglets. Therefore, now we have a total of four weakly charged doublets, whch removes the Wtten anomaly. It s convenent to use the Weyl bass for the fermons n Eq. (3.) and arrange them n a vector that transforms accordng to the fundamental representaton of SU(4). Frst, we wll dene followng left-handed spnors, ~U L;;a V ab " U ;b _ R = V ab " = V ab " ~D L;;a V ab " D _ ;b R ;b U _ R D _ ;b R = V ab " U R ;b; (3.5) = V ab " D R ;b; (3.6) such that we can construct a vector n avor space whch transforms unformly under Lorentz transformatons and gauge transformatons as a left-handed eld n the adjont of SU()TC. We can construct the vector Page 5 of 93

54 CHAPTER 3. MINIMAL WALKING TECHNICOLOR Q A L;;a = U L;;a D L;;a V ab " (U R );b V ab " (D R );b C A U L;;a D L;;a ~U L;;a ~D L;;a C A ; (3.7) where A = ; : : : ; 4 s an SU(4) ndex. The possble knetc terms of the left- and rght-handed Weyl spnors are U y U L U y R U _ R; where all Lorentz ndces are contracted, and we can also construct smlar knetc terms for the techndown. These are nvarant under transformatons of the SU()L or SU()R group for the left- and rght-handed term, respectvely. and (3.8) We can make the theory to a gauge theory by makng the substtuton of the covarant dervatve nstead of the partal dervatve. I.e. we make the D ab ab + g T C A T ;ab, where A are the gauge elds, Tab are the generators of the gauge group, and g T C s the techncolor couplng constant. Therefore, the knetc terms for the left- and rght-handed technup and techndown can be wrtten as follows L K = U y L _;a _ D ab U L;;b + D y L UR y ;a _ Dab U _;b R + Dy R _;a _ D ab ;a _ Dab D _;b R ; D L;;b + (3.9) whch s nvarant under transformatons of the SU()L SU()R symmetry group, and where D ab ab + g T C A T ab s the covarant dervatve. Ths knetc Lagrangan wth the SU() L SU()R symmetry can be wrtten nstead as a knetc Lagrangan wth SU(4) symmetry whch conssts of the Q A L;;a eld (dened n Eq. (3.7)). As shown n eqs. (7.4)-(7.3) n Appendx C-3, such a knetc Lagrangan can be wrtten n terms of the Q vector as follows L K = Q ya L _;a _ D ab Q A L;;b =QyA L; _;a _ D ab Q A L;;b =Q ya L; _;a ab + g T C A T ;ab Q A L;;b: (3.) The second term n the covarant dervatve n Eq. (3.) can be rewrtten by usng that f we have an untary transformaton n Eq. (7.), V T V = (T ) ; (3.) then for V = I such that T = (T ) for every we have that the representaton R s real. If V 6= I, we have that the representaton R s pseudoreal. If such untary matrx does not exst, the representaton R s complex. If T s n the complex representaton n Eq. (7.), then we have not the SU(4) symmetry, because we can not perform the last step n Eq. (7.). We have nstead only the SU ()L SU()R Page 5 of 93

55 CHAPTER 3. MINIMAL WALKING TECHNICOLOR symmetry as n the knetc Lagrangan n Eq. (3.9). In our case, the gauge group s a SU ()TC gauge group, whch s n the pseudoreal representaton, therefore there s a SU(4) symmetry. In the followng we wll derve an expresson of the mass term n terms of the SU(4) vector Q. Assumng the SU(4) symmetry spontaneously breaks to SO(4) or Sp(4). The mass terms have the form U R U L + D R D L = QA L;;aQ B;;a L E AB = QT E Q; (3.) whch s derved n Eq. (7.4) n Appendx C-3, and where the vacuum matrx s E A = We have E + n the mass term f the matrx V ab s symmetrc (V ab = V ba ), and E (V ab = V ba ) as shown n Eqs. (7.4)-(7.) n Appendx C-3. or E C A : (3.3) f t s antsymmetrc We can show that a spontaneous breakng of the global SU(4) symmetry to SO(4) or Sp(4) (for E + respectvely) s drven by the condensate hq T E Q = h U R U L + D R D L. The condensate s namely nvarant under the transformatons Q! gq = exp( T )Q for the unbroken generators,.e. the SO(4) generators for E + and Sp(4) for E, whch are shown n Appendx A. The transformaton of the condensate s Q T E Q =Q T g T E gq = Q T (I + (T ) T )E (I + j T j )Q + O( ) =Q T E Q + Q T (T ) T E + j E T j Q + O( ): (3.4) Thus, the condensate s nvarant f the followng crteron s satsed T T E + E T = : (3.5) These knd of vacua are called techncolor vacua, because the chral symmetry and the EW symmetry of ther condensate break smultaneously. In the composte-hggs vacua whch wll be ntroduced n the next chapter, the EW symmetry s unbroken after the chral symmetry breakng. These vacua are dscussed for rst tme n Refs. [6, 7]. The representaton of SU(4) n Eq. (7.) n Appendx A can be nserted nto the crteron n Eq. (3.5) to show that the condensate s nvarant under SO(4) for + sgn and Sp(4) for sgn, respectvely. By nsertng the S a generators (Eqs. (7.)-(7.3) n Appendx C-3), we obtan (S a ) T E + E S a B B B B and by nsertng the X generators, we get (X ) T E + E X D + D C T C D D A ; a = ; : : : ; 6; (3.6) A ; = ; : : : ; 9: (3.7) Page 53 of 93

56 CHAPTER 3. MINIMAL WALKING TECHNICOLOR We have that B = for the unbroken generators S a when a = ; : : : ; 4, A = for S a when a = 5; 6, D = for the broken generators X when = ; ; 3, and C = for X when = 4; : : : ; 9. Therefore, for E + the relaton n Eq. (3.5) s mantaned for the generators S a but not for X. For E the relaton s mantaned for the generators S a where a = ; : : : ; 4 and for X where = 4; : : : ; 9, whch are the Sp(4) generators. Thus, the condensate n Eq. (3.) s nvarant under SO(4) transformatons for E + nvarant under Sp(4) transformatons for E. Therefore, we use the vacuum matrx E + n ths theory, because the SU(4) symmetry spontaneously breaks to SO(4) drven by the condensate hq T E + Q. Ths leaves us wth nne broken generators wth assocated Goldstone bosons. In Eqs. (7.)-(7.37) n Appendx B s shown that the gauge anomales cancel n the SM. In Eqs. (7.38)-(7.4) n Appendx B s shown that the gauge anomales cancel n MWT, when we have followng generc hypercharge assgnment Y (Q y y + L ) = ; Y (U R; D R ) = ; y ; Y (L y 3y + L ) = 3 ; Y (N 3y R; E R ) = ; ; and (3.8) where the parameter y can be any real value for ths theory, and the electrc charge s Q = T 3 + Y, where T 3 s the weak sospn generator and Y s the hypercharge. If y = =3 then we recover the SM hypercharge assgnment. By usng the matrx notaton of n Eq. (7.96) n Appendx F and the left- and rght-handed Drac spnor and ther adjont n Eqs. (7.6)-(7.9) n Appendx C-3, we can rewrte the followng terms n terms of Drac spnors to n terms of Wayl spnors nstead as follows U L D U L + U R D U R = (U y L ) _;a (U y = U y L _;a _ D ab U L;;b + U y R A D ab A D U U _;b R ;a _ Dab U _;b R : By usng ths rewrtng, the doublets and the snglets of the technfermons n Eq. (3.3) and (3.4), we can rewrte the guage-knetc Lagrangan n Eq. (3.9) to A + L K = Q L D Q L + U R D U R + D R D D R : (3.9) The gauge-knetc terms of the New Leptons have the same form as the technquarks. Thus, we can replace the Hggs sector of the SM wth the MWT Lagrangan L MW T = 4 F a F a + Q L D Q L + U R D U R + D R D D R (3.) + L L D L L + N R D N R + E R D E R ; A where the techncolor eld strength tensor s F a A A a +g TC" abc A b A c, the covarant dervatve Page 54 of 93

57 CHAPTER 3. MINIMAL WALKING TECHNICOLOR of the left-handed technquarks s D ab = + g TC A c " abc g ~ W ~ ab g B Y ab ; (3.) where the A elds are the techn gauge bosons, and W and B are the weak gauge bosons assocated to weak sospn SU()W and the hypercharge U()Y, respectvely. The a matrces are the Paul matrces (Eq. (7.3) n Appendx A), and " abc s the antsymmetrc tensor. For rght-handed technfermons the thrd term n Eq. (3.) contanng the weak nteractons dsappears and for the New Leptons the second term contanng the techncolor nteractons dsappears. The hypercharge generator Y n the last term s replaced wth the approprate hypercharge assgnment n Eq. (3.8). 3. Low Energy Theory for MWT In ths secton, we want to construct the eectve theory for MWT, whch ncludes the composte scalars (e.g. a Hggs scalar), the composte vector bosons and the SM fermons. The eectve theory wll nclude these composte partcle self nteractons and ther nteractons wth the electroweak gauge elds. We wll focus on blnears, because we expect they domnate at low energy. If we have two spn-/ technquarks, then we can construct scalar blenears (spn-) and vector blnears (spn-) from the Q vector n Eq. (3.7) as follows s = : M AB Q A ;aq B;a " ; s = : A ;B A Q A;a _ Q _ ;B;a or (3.) because A ;B A Q C;a _ Q _ ;C;a B A wth A; B = ; : : : ; 4; = +. If we have one spn-/ technquark and one spn- gauge boson, then we can also construct a spn-/ blnear as follows s = = : P A; Q A ;aa a wth A; B = ; : : : ; 4; (3.3) because = +. 3 We start by descrbng the scalar sector, and thereafter we wll descrbe the vector boson sector. In the end of the secton, we descrbe the fermons n the eectve theory and ther Yukawa couplng to the scalar resonances. 3.. Composte Scalars We want to construct a relevant eectve theory for the Hggs sector lke n QCD but nstead at the electroweak scale. Ths eectve theory conssts of a composte Hggs and ts pseudoscalar partner, and nne pseudoscalar Goldstone bosons and ther scalar partners. These composte partcles are assembled The Goldstone bosons s among blnears that domnatng mostly, and for non-gbs we assume that ther masses scale wth the number of the consttuent fermons Page 55 of 93

58 CHAPTER 3. MINIMAL WALKING TECHNICOLOR nto a 4 4 complex matrx M wth the quantum numbers of the rst technquark blnear n Eq. (3.). We can show that ths technquark blnear s symmetrc by swtchng the two Q vectors as follows M AB Q A ;aq B ;b ab " = Q B ;bq A ;a ab " = ( ) Q B ;bq A ;a ba " = Q B ;aq A ;b ab " M BA : Therefore, we need to construct the matrx M such that t s symmetrc by combnng the broken generators and the vacuum matrx E E +. We get that M = h + p a X a E; (3.4) where the X a 's wth a = ; ; 9 are the broken generators of the SU(4) group, whch are lsted n Appendx A. The = v +h eld s a scalar whch may acqure a vev v and the a elds are pseudoscalars. Ths M matrx transforms under SU(4) group accordng to M! gmg T ; g SU(4); (3.5) where the SU(4) element can be expanded as g = exp( a T a ) ' + a T a wth a = ; : : : ; 5 for nntesmal small a phases. We can make a SU(4) transformaton of the M matrx M! M = gmg T = M + a [T a M + MT at ] + O( ); (3.6) Thus, the M matrx s nvarant under the SU(4) generators T a, f T a M + MT at '. The rst term of the M matrx s the only term that s nvarant under SO(4) transformatons, because the crtera S a E + ES at = are mantaned whle X E + EX T 6= accordng to Eq. (3.6) and Eq. (3.7). Therefore, the broken generators do not leave the vacuum expectaton value (VEV) of M nvarant hm v = E: (3.7) The second term of the M matrx s nvarant under the SO(4) transformaton but not nvarant under SU(4). Ths s shown n Eq. (7.4) n Appendx C-3, whch gves that S b X a E + X a ES bt = : Therefore, the M matrx n Eq. (3.4) s not a representaton of SU(4). We can transform the M matrx n Eq. (3.4) where t s wrtten n terms of the elds and a as follows M!gMg T ' + b T b h + p a X a E + b T bt = + ( a T a + E a T at E) p + a + c T c a c b X b T ct EX a X a E: (3.8) Therefore, we need to add an extra psudoscalar,, and nne extra scalars, ~ a, to make the M matrx closed under SU(4) transformatons. Thus, we have that the M matrx s Page 56 of 93

59 CHAPTER 3. MINIMAL WALKING TECHNICOLOR M = + p + ( a + ~ a )X E: a (3.9) Ths M matrx s a representaton of SU(4), whch conssts of degrees of freedom or complex degrees of freedom. We have the elds:,, a and ~ a wth a = ; : : : ; 9. The connecton between the composte scalars n Eq. (3.9) and the underlyng technquarks can be derved by observng that the elements of the matrx M transform lke a technquark blnears as the rst blnear n Eq. (3.) M AB Q AQ B " wth A; B = ; : : : ; 4; (3.3) By usng ths connecton, the SU(4) generator matrces n Appendx A and the spnor blnears n Appendx F, we have related the scalar elds to the wavefunctons of the technquark bound states. The results are shown n Eq. (7.97) and Eq. (7.98) n Appendx F. Ths gves for the technmesons, whch are composed of one technquark and one ant-technquark, the followng charge states v + H UU + DD; ( U 5 U + D 5 D); A ~ 3 UU DD; 3 ( U 5 U D 5 D); A + ~ ~ p DU; + p D 5 U; (3.3) A ~ + ~ p UD; + p U 5 D: For the technbaryons made up of two technquarks (wth two derent colors), we have that UU U T CU; DD D T CD; UD 8 + p 9 U T CD; ~ UU ~ 4 + ~ 5 + ~ 6 + ~ 7 U T C 5 U; ~ DD ~ 4 + ~ 5 ~ 6 ~ 7 D T C 5 D; ~ UD ~ 8 + ~ 9 p U T C 5 D; (3.3) where U = (U L; ; U R _ )T and D = (D L; ; D R _ )T are the up- and down-technquark, and C s the charge conjugaton matrx (shown n Eq. (7.96)). To these technbaryon charge states we have also ther correspondng charge conjugate states, e.g. nstead of UU we have UU by makng the substtuton n Eq. (7.5) n Appendx F. As shown n Appendx F, the elements of the M matrx can be rewrtten n terms of these technmeson and technbaryon charge states as follows Page 57 of 93

60 CHAPTER 3. MINIMAL WALKING TECHNICOLOR M = UU + ~ UD+ UD ~ UU p + + +A + +A + p UD+ p UD ~ DD + ~ +A DD p + A + + +A + +A + p + A p+a UU + ~ UD + ~ p UD UU UD + ~ UD p DD + ~ DD C A : (3.33) The electroweak subgroup can be embedded n SU(4). The generators S a wth a = ; ; 3 n Appendx A form the vectoral SU() subgroup of SU(4), denoted SU()V, and the generator S 4 subgroup. forms a U()V These two subgroups together wth the broken generators X a wth a = ; ; 3 generate a SU()L SU()R U()V subgroup of SU(4). Ths can be seen by changng generator bass (S a ; X a ) to L a Sa + X a p = S 4 = I a = A ; R at Sa X a p at = A ; A ; (3.34) wth a = ; ; 3. By gaugng SU()L (dentfyng t wth SU()W) and U()Y SU()R U()V, the electroweak gauge group SU()W U()Y s obtaned, where p Y = R 3T + Y V S 4 ; (3.35) and Y V s the U()V charge.from the general gauge anomaly free hypercharge assgnment n Eq. (3.8), we see that Y V = y for the technquarks, and Y V = 3y for the New Leptons, because Y Q L; = Y L L; = Y V U L; Y V D L; (Y V + )U R; (Y V )U R; Y V N L; Y V D E; (Y V + )N R; (Y V )E R; C A C A = = yu L; yd L; (y + )U R; (y )D R; 3yN L; 3yE L; C A ( 3y + )N R; ( 3y )E R; and (3.36) C A : (3.37) When SU(4) spontaneously breaks to SO(4), then the global subgroup SU()L SU()R breaks to SU()V SU ()L+R as seen from Eq. (3.34) where the X a are broken. The consequence s that the electroweak gauge group breaks to U()Q, where Q = p p S 3 + Y V S 4 : (3.38) In summary, the global subgroup breakng pattern s SU()L SU()R U()V! SU()V U()V (as n two avor QCD). The resultng EW symmetry breakng pattern s the coset SU ()W U()Y! U()Q. The SU()V group acts as the custodal sospn as n the SM, whch s entrely contaned n the unbroken Page 58 of 93

61 CHAPTER 3. MINIMAL WALKING TECHNICOLOR SO(4) group. Ths ensures that the parameter n Eq. (.7) s equal to one at tree-level. The gaugng of the electroweak symmetry breaks explctly the SU(4) symmetry group down to SU()L SU()R U()V (the gaugng of SU()L gves SU()W and the gaugng of the rest gves U()Y SU()R U()V), whle the spontaneous symmetry breakng leaves a SO(4) subgroup nvarant. Therefore, the remanng unbroken group s SU()V U()V as smple llustrated n Fgure 3.. The gaugng of ths group gves U()Q SU()V U()V. Here s the U()Q group s the symmetry group whch s assocated to the electromagnetsm, whle the U()V symmetry leads to the conservaton of the technbaryon number. SU(4) SU () L SU () R U () V SU () V U () V SO(4) Fgure 3.: Spontaneous breakng from SU(4) to SO(4) due to dynamcs and explct breakng from SU(4) to SU()L SU()R U()V due to EW gaugng. By usng Eq. (3.38), we can calculate the charges of the technmesons and technbaryons. Frstly, we wll nd the charges of the elements of the Q vector n Eq. (3.7). In ths case we have that Y V = y because the charge operator Q works on technquarks. We have that p QQ A L; ps = 4 + Y V S 4 Q L; = ( + y)u L; ( + y)d L; ( y)u R; ( y)d R; C A : (3.39) By addng the two charges for the two technquarks we get the charges of the elements of the M matrx, whch are M AB Q L;AQ L;B; ) Q AB = + y y y + y y y y y C A wth A; B = ; : : : ; 4: (3.4) In Table 3. the scalars are classed accordng to the unbroken group U()V U()Q wth the U()V charge and the U()Q charge, whch are llustrated as the unbroken symmetry group n Fgure 3.. Three of the nne physcal degrees of freedom are eaten up by the longtudnal components of the SM gauge Page 59 of 93

62 CHAPTER 3. MINIMAL WALKING TECHNICOLOR bosons, whle the remanng sx Goldstone bosons carry technbaryon number whch are denoted by UU, DD, UD and ther charge conjugated states. Because these GBs carry technbaryon number, we refer to these states as technbaryons. Feld U () V charge U()Q charge Lnear Combnaton W + L / + + p W L / - + p Z L / 3 UU / ~ UU + y / ~ 4 + ~ 5 + ~ 6 + ~ 7 DD / ~ DD + + y / ~ 4 + ~ 5 ~ 6 ~ 7 UD / ~ UD + y p / ~ 8 + ~ 9 p y UU / ~ y UU - y / ~ 4 ~ 5 + ~ 6 ~ 7 y DD / ~ y DD - y / ~ 4 ~ 5 ~ 6 + ~ 7 y UD / ~ y UD - y 8 9 p / ~ 8 ~ 9 p Table 3.: Classcaton of the Goldstone bosons accordng to the unbroken global group U ()V and the unbroken gauge group U()Q SU()V U()V. In the followng we wll show that the elds A ; and ; are trplets under the custodal symmetry SU()V, whle the elds and are snglets. We know from Eq. (3.33) that the M matrx can be wrtten n the symmetrc form M A B T B C A ; (3.4) where A, B and C are matrces. We have the elements of the SU()L and SU()R are wrtten as g 4 4 L =e a L a = (g 4 4 R ) =e a R at ea e a at A g L A ; A g R A ; (3.4) where L a and R at are the generators n Eq. (3.34), and g R = (exp(a a )) = exp( a a ) = exp( a at ). Thus, the M matrx transforms under the SU()L SU()R symmetry group as where The B matrx can be wrtten n the form M! gmg T g LAgL T g R BT gl T g 4 4 = g 4 4 L + (g 4 4 R ) = g L Bg y R g R Cgy g L g R A ; (3.43) A : (3.44) Page 6 of 93

63 CHAPTER 3. MINIMAL WALKING TECHNICOLOR B = + + ~ + : (3.45) The chral symmetry breakng SU(4)! SO(4) gves rse to that the group SU()L SU()R U()V breaks to SU()V U()V, where SU()V s the custodal symmetry group,.e. g L = g R = g V. Thus, the rst term n B s nvarant under SU()V transformatons as follows B ()! g V B () g y V = + a a + b b + O( ) whle the second term n B transforms as follows = + + O( ); B ()! g V B () g y V = + a a ~ + b b + O( ) ~ + = + a ~ + [ a ; ] + O( ): (3.46) (3.47) The elds and ~ mx wth each other by transformng them under SU()V, respectvely. Thus, the elds A ; and ; n Eq. (3.3) form each a trplet under SU()V. However, the elds and are both a snglet under SU()V. We wll now construct an eectve Lagrangan wth the M matrx as n QCD. The electroweak covarant dervatve for the M matrx has the form D M M g[g (Y V )M + MG T (Y V )]; (3.48) where we have Y V = y because we have to take the U()Y charge of the technquark consttuents n the M matrx as shown n Eq. (3.3), and gg (Y V ) =gw a L a + g B Y =gw a L a + g B R 3T + py V S 4 : (3.49) Under electroweak gauge transformatons, we have that M transforms as follows M (x)! u(x; y)m (x)u T (x; y); (3.5) where h u(x; Y V ) = exp a (x)l a + (x) R 3T + py V S 4 ; (3.5) and Y V = y because the M matrx conssts of technquarks. We can construct an eectve Lagrangan at low energy. The eectve Lagrangan must respect the global symmetres as the underlyng Lagrangan. Furthermore, t must be nvarant under the electroweak gauge transformatons and CP transformatons. The new Hggs Lagrangan s L Hggs = Tr[D MD M y ] V (M ) + L ETC ; (3.5) where the potental s Page 6 of 93

64 CHAPTER 3. MINIMAL WALKING TECHNICOLOR V (M ) = m Tr[MMy ] + 4 Tr[MMy ] + Tr[MM y MM y ] [DetM + DetM y ]; (3.53) and L ETC s all the terms whch are generated by the extended techncolor nteractons (ETC) and not by the chral symmetry breakng sector. We can not use the countng scheme n dervatves as n Eq. (.3) for the rst three terms n the potental, but as dscussed below Eq. (.3) we can gnore many-partcle vertces. Frstly, t s hard to produce and therefore not nterestng to consder. Secondly, at a gven number of external lnes n the vertex then the energy would be above the scale where the composte partcles would fall apart. All the terms n the Lagrangan L Hggs are nvarant under a global transformatons, gauge transformatons and CP transformatons. In Appendx G the dscrete transformatons (party, charge conjugaton and CP transformatons) of spnors, the Q vector and the M matrx are derved. We can notce that the dertermnant terms explctly break the U()A symmetry, whch gve mass to. Ths exctaton would otherwse be a massless Goldstone boson. Three of the nne Goldstone bosons assocated wth the nne broken generators S a become longtudnal degrees of freedom of the massve weak gauge bosons. The last sx Goldstone bosons wll acheve a mass from the extended techncolor nteractons (ETC) and the electroweak nteractons. Accordng to Eq. (6) n Ref. [], the ETC nteracton terms can be wrtten as follows L ET C = m ET C Tr[MBM y B + MM y ] + ; (3.54) 4 where B p S 4 and the extra terms could be hgher dmensonal operators. ETC terms generate also the masses of the SM fermons as explaned later. Fnally, we can determne from the Lagrangan n Eq. (3.5) the vacuum expectaton value (VEV) of the composte Hggs and the masses of the composte scalars n terms of the model parameters. By usng the Mathematca, the vacuum expectaton value (VEV) of the Hggs canddate s v = h = m + + : (3.55) By usng the Mathematca, we have that the Hggs mass term and therefore the Hggs mass s MH H = 3 m v 3 v + 3 v H = 3 m m H = m H ) M H = m : (3.56) The same procedure s carred out for the remanng composte technmesons and technbaryons. The masses of the remanng technmesons are M m = v v 3 v = ( + ) v v 3 v = v ) M = 4v ; (3.57) Page 6 of 93

65 CHAPTER 3. MINIMAL WALKING TECHNICOLOR whch reveves ts mass from the explctly breakng of the U()A symmetry by the determnant terms wth the couplng constant n Eq. (3.53), and M A A+ A + A A + M A = ~ + ~ = (v + v ) ~ + ~ ) M A = v ( + ); (3.58) M A A A M A = ~ 3 ~ 3 = v ( + ) ~ 3 ~ 3 ) M A = v ( + ; ) (3.59) and the three pseudoscalar mesons and are massless, and they correspond to the three massless Goldstone bosons whch are eaten by the longtudnal degrees of freedom of the massve W and Z boson. The remanng sx uneaten Goldstone bosons are the technbaryons, whch acqure tree-level degenerate masses by the not speced ETC nteractons M UU UU UU M DD DD DD = 4 M UU M DD + ( ) = 4 m ET C ( ) ) M UU = M DD = m ET C and (3.6) M M UD UD UD UD = ) M UD = m ET C; ( 8 9 )( ) = M UD ( ) = m ET C ( ) (3.6) The degenerate mass of the remanng technbaryons s = M ~ UU ~ U U ~ UU + ~ UU ~ U U M ~ UU + M ~ DD ( ~ 4 + ~ 5 + ~ 6 + ~ 7 ) M ~ DD ~ D D ~ DD + ~ DD ~ D D = m ET C ( ~ 4 + ~ 5 + ~ 6 + ~ 7 ) v ( + )( ~ 4 + ~ 5 + ~ 6 + ~ 7 ) ) M UU ~ = M DD ~ = m ET C + v ( + ); (3.6) and M ~ UD ( ~ U D ~ U D + ~ UD ~ M U D ) = ~ UD = m ET C ( ~ 8 ) + ( ~ 9 ) v ( + ) ( ~ 8 ) + ( ~ 9 ) ( ~ 8 ~ 9 )( ~ 8 + M ~ 9 UD ~ ) = ( ~ 8 ) + ( ~ 9 ) ) M ~ UD = m ET C + v ( + ): (3.63) 3.. Composte Vector Bosons The composte vector bosons of ths theory are convenently descrbed by A = A a T a ; (3.64) where T a are the SU(4) generators wth T a = S a for a = ; : : : ; 6, and T a+6 = X a for a = ; : : : ; 9 n Page 63 of 93

66 CHAPTER 3. MINIMAL WALKING TECHNICOLOR Appendx A. We have that A transforms under an SU(4) transformaton as follows A! ga g y ; where g SU(4): (3.65) Accordng to the tracelessness of the matrx A n Eq. (3.64) and the SU(4) transformaton of the matrx n Eq. (3.65), ths gves a connecton of ths matrx A wth the two lower technquark blnears n Eq. (3.) A ;B A Q A _ Qy _ ;B 4 Q C _ Qy _ ;C B A wth A; B; C = ; : : : ; 4; (3.66) whch s traceless because Tr(A ;B A ) = Q A Q ya 4 A A Q C Q yc =, and t transforms as Eq. (3.65), because A ;B A = Q A Q yb 4 Q C Q yc B A! g C AQ C Q yd g yb D 4 gd C Q D Q ye g yc E B A = gc AQ C Q yd g yb D 4 Q E Q ye B A = gaq C C Q yd g yb D 4 gc AQ E Q ye C D g yb D = gc AA ;D C gyb D : (3.67) In Appendx F, the relatons between the charge egenstates and the wavefunctons of the composte vector mesons are derved, whch are v A 3 U U D D; a A 9 U 5 U D 5 D; v + A A p D U; a + A7 A 8 p D 5 U; v A + A p U D; a A7 + A 8 p U 5 D; (3.68) v 4 A 4 U U + D D; and for the vector baryons we have that x UU A + A + A + A 3 U T C 5 U; x DD A + A A A 3 D T C 5 D; x UD A4 + A 5 p D T C 5 U; (3.69) s UD A6 A 5 p U T C D: In Eq. (7.3) n Appendx F, we have also derved the A matrx whch s dened n Eq. (3.64) wth the vector technmesons and technbaryons n Eq. (7.6) and (7.7) A = a +v +v 4 a +v a + +v + a p v +v 4 x x puu UD s UD x UD +s UD x p UU x UD s UD a v v 4 p x pdd a + v + x UD +s UD x p DD a v a +v p v 4 C A : (3.7) Page 64 of 93

67 CHAPTER 3. MINIMAL WALKING TECHNICOLOR The knetc Lagrangan s L kn = Tr[ W ~ ~ W ] B B Tr[F F ] + m ATr[C C ]; (3.7) 4 where W ~ and B are the ordnary eld strength tensors for the electroweak gauge elds and F s the new eld strength tensor for the new SU(4) vector bosons, whch s F A ~g[a ; A ]; (3.7) and we have dened the vector eld C as follows C A g ~g G (y); (3.73) where G (y) s the vector eld n Eq. (3.49) wth Y V = y. The tensor W ~ are not yet the SM weak trplets. They mx wth the composte vector bosons to form mass egenstates whch correspondng to the ordnary W and Z bosons. The vector eld C transforms as follows C (x)! u(x; y)c (x)u(x; y) y ; (3.74) where u(x; Y V ) s gven by Eq. (3.5). Ths vector eld transform lke a gauge eld wth the excepton of the extra term wth u@ u y n gauge transformatons. The terms n the Lagrangan are not only knetc ones, because t contans self-nteracton terms and one mass term. The mass term s gauge nvarant, whch gves a degenerate mass m A to all the composte bosons, whle leavng the gauge bosons massless. The gauge bosons acqure ther mass from the covarant dervatve term of the scalar matrx M n Eq. (3.5) after spontaneous symmetry breakng. We can construct an eectve Lagrangan where the C elds couple to the M matrx up to dmenson four operators. The eectve Lagrangan can be wrtten as L M C =~g r Tr[C C MM y ] + ~g r Tr[C MC T M y ] + ~gr 3 Tr C (M (D M ) y (D M )M y ) + ~g str[c C ]Tr[MM y ]; (3.75) where the dmensonless parameters r, r, r 3 and s are the derent strength of the nteractons between the composte scalars and vectors n unts of ~g, therefore they are expected to be of order one. The terms n the eectve Lagrangan are global SU(4) nvarant, gauge nvarant and CP nvarant Fermons n the Eectve Theory The fermonc content of the eectve theory conssts of the SM quarks and leptons, a composte technquarktechngluon doublet, and the New Lepton doublet whch s ntroduced to cure the Wtten anomaly. We want to extend the SU(4) symmetry to the ordnary quarks and leptons. We arrange the SU ()W doublets n SU(4) multplets as we have done for the technquarks n Eq. (3.7). For the SM quarks and leptons, we ntroduce the four component vectors Page 65 of 93

68 CHAPTER 3. MINIMAL WALKING TECHNICOLOR q A; L; = u L; d L; " (u R ); " (d R ); C A and l A; L; = L; e L; " ( R ); " (e R ); C A ; (3.76) where s the generaton ndex. To have ths extended SU(4) symmetry then we need to ntroduce a rght-handed neutrno for each generaton. In addton to these SM SU(4) multplets, we have an multplet for the New Leptons and technquark-techngluon bound state, L A L; = N L; E L; " (N R ) " (E R ) C A and ~ Q A L; = QyA; _ L _ A = ~U L; ~D L; " ( ~ U R ) " ( ~ D R ) We can wrte the fermon Lagrangan wth a SU(4) global symmetry as follows C A : (3.77) L fermon =q y D q + ly D l + Ly D L + ~ Q y D ~ Q + x ~ Q y C ~ Q ; (3.78) where the electroweak covarant dervatve for the fermon elds can be wrtten as D gg (Y V ); (3.79) where G (Y V ) s gven n Eq. (3.49), and the vector eld C s dened n Eq. (3.74). The U()V charge s Y V = =3 for the SM quarks, Y V = for the SM leptons, Y V = 3y for the New Lepton doublet, and Y V = y for the technquark-techngluon bound state. The rst four terms n the Lagrangan are the knetc terms of the fermons lke the Lagrangan term n Eq. (3.). The last term n the Lagrangan whch couples Q ~ to C s always allowed, because the term s nvarant under electroweak gauge transformatons for any Y V = y. Any SU(4) fermon multplet transforms as follows (x)! u(x; Y V ) (x); (3.8) and C transforms as follows C (x)! u(x; y)c (x)u(x; y) y ; (3.8) where u(x; Y V ) s gven n Eq. (3.5). We have that Y V = y for C due to the fact that the composte vectors are bult out of technquark blnears. Thus, we have that the term y _ C transforms lke y _ C! y u(x; Y V ) y _ u(x; y)c u(x; y) y u(x; Y V ) ; (3.8) and therefore the term s only nvarant f Y V = y. For y 6= =3 and y 6=, we have that the term s only nvarant for = ~ Q (the last term n Eq. (3.78)). For y = =3 or y =, we have that the term s not only nvarant for = ~ Q, but also for ether = q or = l, respectvely. Page 66 of 93

69 CHAPTER 3. MINIMAL WALKING TECHNICOLOR 3..4 Yukawa Interactons In ths secton we wll provde masses to ordnary fermons. There are many extensons of techncolor to provde the fermon masses. One way could be to use another strongly coupled gauge dynamcs or ntroduce new fundamental scalars. Such a model s called an extended techncolor (ETC) theory, whch we dscuss later. In ths secton we are smply couple the fermons to our low energy eectve Hggs to keep the number of elds mnmal. Ths s done by wrtng Yukawa nteractons whch couple the SM fermons to the matrx M. These Yukawa terms are dependng on the value of y for the technquarks. We denote as ether q or l. We can wrte the Yukawa term T M + h.c.; (3.83) whch s electroweak gauge nvarant, when the U()V charges of and the technquark multplets Q a are the same. The Yukawa term s nvarant for Y V = y, because t transforms (accordng to Eq. (3.5) and (3.8)) as follows T M! T u(x; Y V ) T u(x; y) M u(x; y) y u(x; Y V ) ; (3.84) where u(x; y) y u(x; y) = u(x; y)u(x; y) y =, and therefore we have that u(x; y) T u(x; y) =. Otherwse, f the U()V charges of and Q a are derent, then we can only wrte a gauge nvarant Yukawa term wth the o-dagonal M (contans only the Hggs boson and the Goldstone bosons),.e. M o + + +A + +A + p + A + + +A + +A + p p+a + A p+a C A : (3.85) Ths Yukawa term s wrtten as T M o + h.c.; (3.86) because the U()V charge of the M o s zero, snce S 4 M o + M o S 4T = (3.87) accordng to Eq. (7.5) n Appendx C-3. Therefore, the U()V charges of T and need to cancel each other n Eq. (3.86). The Yukawa term n Eq. (3.86) s the only vable for the New Leptons, because the correspondng U()V charge s derent from the charge of the technquark multplets Q a (Y V = 3y 6= y). For the SM quarks, the Eq. (3.83) contans quark-quark terms whch are not color snglets. Therefore, the only vable Yukawa term for the ordnary quarks s the term n Eq. (3.86). However, we notce that the Yukawa terms n Eq. (3.83) and (3.86) are not phenomenologcally vable yet, because the SU()L subgroup of SU(4) are unbroken and there are no dstngush between the up-type and down-type fermons n these Yukawa terms. Therefore, we break the SU()L symmetry to U()R by Page 67 of 93

70 CHAPTER 3. MINIMAL WALKING TECHNICOLOR usng the projecton operators (as done n eq. (.77) when we talked about custodal symmetry) P U + 3 Thus, we replace Eq. (3.83) and Eq. (3.86) wth A and PD 3 A (3.88) T (PU M P U ) T (PD M P D ) + h.c.; (3.89) and T (PU M op U ) T (PD M op D ) + h.c.: (3.9) In the next, we would wrte the Yukawa nteractons for two derent cases, y = and y 6=. For y =, we can form gauge nvarant Yukawa terms wth the SM leptons and the full M matrx. Therefore, the Yukawa Lagrangan for ths case s L Yukawa = y j u q T (P U M op U )q j y j d qt (P D M op D )q j y j l T (P U M P U )l j y j e l T (P D M P D )l j y N L T (P U M op U )L y E L T (P D M op D )L (3.9) y ~U ~ Q T (P U M P U ) ~ Q y ~D ~ Q T (P D M P D ) ~ Q + h.c.; where yu j, y j d, yj and ye j are arbtrary complex matrces, and y N, y E, y ~U and y ~D are complex numbers. For y 6=, we can only form gauge nvarant Yukawa terms wth the SM fermons and the o-dagonal M matrx L Yukawa = y j u q T (P U M op U )q j y j d qt (P D M op D )q j 3.3 Extended Techncolor Models y j l T (P U M op U )l j y j e l T (P D M op D )l j y N L T (P U M op U )L y E L T (P D M op D )L y ~U ~ Q T (P U M P U ) ~ Q y ~D ~ Q T (P D M P D ) ~ Q + h.c.: (3.9) In a techncolor model we need to ncorporate a mechansm that generates quark and lepton masses, the varous weak mxng angles, and the CP-volaton. Thus, we have that the quarks and leptons of the SM need to couple to the technquark condensate. In addton, there must be a mechansm that volates the technbaryon quantum number, because the technquarks must be able to decay, snce there are no stable technbaryons observed n the unverse. A popular way to solve these requrements s to extend the Techncolor gauge nteractons wth some extended gauge bosons, whch couple both to SM fermons and technquarks. These extended nteractons are part of a large gauge group G ETC whch breaks down to the techncolor subgroup at an energy ETC. Ths energy scale ETC s above the scale TC at whch the techncolor couplng becomes strong. From a hgh-energy theory based on a master gauge group G ETC, t s possble to obtan a low-energy Page 68 of 93

71 CHAPTER 3. MINIMAL WALKING TECHNICOLOR theory where the only survvng gauge groups are those of techncolor and the SM. The master gauge group G ETC undergoes a symmetry breakng at the scale ETC, where t breaks down to the techncolor gauge group G TC as follows G ETC! G TC G SM at ETC; (3.93) where the remanng groups n addton to G TC are nclude the full Standard Model G SM = SU(3)C SU()W U()Y. In the new nteractons are requred couplngs of technquarks Q L;R nto the SM quarks and leptons L;R (q L;R and l L;R ) wth the currents of the form Q L;R L;R, whch couple to the new ETC gauge bosons. The full theory wth the master gauge group G ET C contans the desred currents of the form, Q and Q Q. A smple example could be that the Techncolor group SU(N TC ) s embedded nto a larger ETC group SU (N ETC ), where of course we have that N ETC > N TC. At low energy scale. ETC, we have that the heavy ETC bosons, whch exchange from the currents correspondng to the broken ETC generators T a, produces three types of eectve contact nteractons between the technquarks and the SM fermons, whch (cf. page 59 n Ref. [5]) are ab Q L T a Q R R T b L ETC + ab QT a QQT b Q ETC + ab LT a R R T b L ETC + : : : ; (3.94) where the ab, ab and ab are coecents that are contracted wth generator ndces, where ther structure depends upon the constructon of the ETC theory. Energy Scale Q L q R g L Z Q L q L + g R Z Q R q R g L Z M Z p g R ET C M Z M Z q L Q R Q L Q R g L g R M Z q L Q L Q Rq R g L g R M Z v EW hqq =3 4f TC q L q R q R gl g R v EW M Z q L M o q R q M o q L Fgure 3.: The varous symmetry breakngs from a ETC gauge symmetry G ETC for ETC gauge bosons Z to G SM after EW spontaneous symmetry breakng, whch produce the masses of the SM fermons. Page 69 of 93

72 CHAPTER 3. MINIMAL WALKING TECHNICOLOR The -term n Eq. (3.94) s responsble for gvng masses for the SM fermons m f hq LQ R TC ETC hq = g L g L Q R TC R ; (3.95) METC where g L and g R are the ETC gauge couplngs to the left- and rght-handed fermons, respectvely, M ETC = ETC= p g L g R s the mass of the ETC gauge boson, and hq L Q R TC s the technquark condensate evaluated at the TC scale TC. An llustraton of the varous symmetry breakngs of an ETC theory s shown n Fgure 3., where the TC and EW breakng happen at same energy scale TC v EW as n the SO(4) Mnmal Walkng Techncolor (MWT). At energes over ETC the ETC gauge bosons nteract wth both the technfermons and the fermons wth the ETC gauge couplngs g L;R. When the energy s lowered below ETC, then the ETC gauge propagator can be ntegrated out such that we have an eectve four fermon vertex. Fnally, when the energy s lowered below the TC scale, then the technquarks condense and we get the condensate hq L Q R TC. In ths specal case, we have that the condensate s hq L Q R TC = 4vEW 3. Overall, we have the followng symmetry breakng pattern: g L ZQ L q L + g R ZQ R M q Z R! ETC v EW! 4f 3 ETC q L Q LQ R q R = g Lg R M Z q L Q LQ R q R q L q R = g Lg R v 3 EW M Z q L q R; (3.96) where the Yukawa couplngs are q = g Lg R v EW M Z : (3.97) The form of the matrx M o s shown n Eq. (3.85), where = v EW + h. Therefore for ths specal case the masses of the SM fermons n MWT are Ths mass formula can easly be generalzed to the Eq. (3.95). m f g L g R v 3 EW M Z ; (3.98) The fermon masses can also be produced by new heavy scalar elds H whch nteract wth the technquark and the SM fermons wth the Yukawa couplng Q and q, respectvely. The varous symmetry breakngs are shown n Fgure 3.3, where the new scalar propagators are ntegrated out, when the energy s lowered below the ETC scale, whch gve the -term n Eq. (3.94. When the energy s lowered below TC scale, we get the Yukawa terms shown n the gure, and thus the Yukawa terms n ether Eq. (3.9) or Eq. (3.9). Overall, we have the followng symmetry breakngs for new heavy scalars: EM Q Q L HQ R + R q L Hq H R! ETC Ev! 4f 3 ETC Q LQ R q Lq Q q R = Q M LQ H R q Lq R q Lq R = Q q v 3 EW M H q Lq R ; (3.99) where the Yukawa couplngs are Q q vew q = : (3.) MH Therefore for ths specal case wth new heavy scalars the masses of the SM fermons n MWT are Page 7 of 93

73 CHAPTER 3. MINIMAL WALKING TECHNICOLOR m f Q q vew 3 : (3.) MH By extendng the Techncolor theory by addng heavy gauge bosons or new heavy scalars, we have moved the naturalness problem further up the energy scale, because we have a new scalar whch mass should be ne-tuned. Therefore, we have only reduced the naturalness problem and not removed t. It s also the case n the ETC theores wth new gauge bosons, because we need new scalar elds lke the Hggs boson n SM to make the gauge bosons massve after spontaneous symmetry breakng. Addtonally, the -term contrbutes also to mxng angles between quarks and leptons,.e. t contrbutes to the parameters of the CKM and the PMNS matrx. Energy Scale Q L q L Q Q L HQ R + q q L Hq R Q H M H p q ET C M H M H Q R q R Q L q R Q q M H Q q M H Q L Q Rq L q R v EW hqq =3 4f TC Q R q L q L Q q v EW M H q L M oq R q M o q R Fgure 3.3: The varous symmetry breakngs from a ETC gauge symmetry G ETC for heavy scalars H to G SM after EW spontaneous symmetry breakng, whch produce the masses of the SM fermons. The mass herarchy between the generatons of the fermons can be acheved breakng G ETC n several steps as follows G ETC! G n! G n!! G! G TC G SM : (3.) Some of the ETC gauge bosons become massve durng the every step, whch gves derent ETC scales ETC M ETC. Thus, ths scenaro produces derent fermon masses as desred accordng to Eq. (3.95). Ths way to produce the fermon mass herarchy s called tumblng. The -term n Eq. (3.94) can nduce masses to the pseudo-goldstone bosons (pngbs). The upper dagram n Fgure 3.4 shows how the ETC propagator s ntegrated out for energes ETC, such Page 7 of 93

74 CHAPTER 3. MINIMAL WALKING TECHNICOLOR that we obtan the four-technfermon operators. These four-technfermon terms can potentally solve a problem that the masses of the PNGBs are too small that we have not observed them. Ths mechansm can elevate the masses of these lght PNGBs to larger values whch are more consstent wth the experments. For example n the SO(4) MWT the sx pngbs (the pngbs whch are not become the longtudnal degrees of freedom of the weak gauge bosons) acheve ther masses from these -terms n Eq. (3.54). The ETC terms n Eq. (3.54) consst both of two M matrces,.e. these terms are four technfermon vertces as the -terms. Thus, ther masses are M UU = M DD = M UD = m ETC and the same mass for ther charge conjugated elds. Fnally, the -term n Eq. (3.94) generates Flavor-changng neutral current (FCNC) contrbutons whch exclude the possblty of generatng large fermon masses n these ETC models. The lower dagram n Fgure 3.4 shows how the ETC propagator s ntegrated out for energes ETC, such that we obtan the four-fermon operators. Q L q L Q L Q R Q H q M H p M H Q M H Q R q R Q R Q L q L q L q L q R q H q M H p M H q M H q R q R q R q L Fgure 3.4: The upper dagrams are the ETC symmetry breakng whch gves a -term where the heavy scalar H propagator s ntegrated out. The lower dagrams are the ETC symmetry breakng whch gves a -term. For example a process lke (s 5 d)(s 5 d) ETC (3.3) s nduced. Ths new contrbuton causes S = FCNC nteractons whch gve a contrbuton to the well-measured K L K S mass derence (short-lved K S (CP = ) and long-lved K L (CP = +) weak egenstate). Ths s an ndrect way to measure of CP volaton due to the mxng of the neutral kaons K and ts antpartcle K, because the K and K has the quark content sd and ds, respectvely, then the four-fermon term n Eq. (3.3) contrbutes to ths mxng and thus the CP volaton. Ths -term yelds the contrbuton to the mass derence (accordng to Eq. (3.98) n Ref. [5]) m m K f K m K ETC. 4 ; (3.4) where f K s the kaon decay constant, m K s the kaon mass, and we expect that sn C realstc model. Therefore, we obtan the lower constrant on the ETC scale n a ETC & 3 TeV: (3.5) Page 7 of 93

75 CHAPTER 3. MINIMAL WALKING TECHNICOLOR where f K MeV and m K 5 MeV. Applyng ths bound and assumng yelds an upper bound on the masses of the SM fermons, whch s m f. MeV: (3.6) Thus, t s already problematc to produce the mass of the charm quark wth ths ETC model. Ths problem can maybe be allevated by the couplng of the techncolor model s walkng n an energy wndow as explaned n the next secton. 3.4 Walkng Techncolor There are problems n buldng models wth fermons there are heavy enough and models wth sucently suppressed avor-changng neutral currents (FCNCs). The ETC models n prevous secton produce not the observed quark and lepton masses. In ths secton we attempt to deal wth these dcultes wth walkng techncolor. The Lagrangan of such a theory has the form L = Q D Q Tr[G G ]; (3.7) 4 where Q are the technquarks and G s the eld strength tensor of the techngluons. Let the technquarks be n the fundamental representaton of SU(N ) as the quarks n QCD. The last term of the Lagrangan s the Yang-Mlls theory, such a theory conssts not of quarks. There s stll connement n such a theory, because at low energes there can be created glueballs, whch s a hypothetcal composte partcle consstng solely of gluon partcles. The -functon for the couplng g (from Eq. (.6.34) n Ref. [3]) log = g 3 (4) + g 5 (4) + g 7 4 (4) + 6 O(g9 ); (3.8) where = 4 T R C A; 3 3 = 34 3 C A + 3 C AT R + 4C R T R ; = C3 A C AT R 58 7 C AT R C AC R T R 44 9 C RT R C RT R : The Casmr operators C A, C R, and the Dynkn ndex T R are dened as follows X a;b X N a= f abc f abd = C A cd ; X j T a jt a jk = C R k ; (3.9) (3.) Tr(T a T b ) = T R ab ; Page 73 of 93

76 CHAPTER 3. MINIMAL WALKING TECHNICOLOR respectvely. For the fundamental representaton of SU (N C ), we have that C A = N C, C R = C F = (N C )=N C and T R = T F N f = N f =, where N f and N C are the number of fermons and colors, respectvely. Nf g QED-lke >, > g g g Banks-Zaks FP g <, >, > g g Walkng Techncolor g <, >, < g <, < QCD-lke g NC Fgure 3.5: Schematc presentaton of the derent scenaros for the RG evoluton of the gauge couplng g and ther -functons n the N f - N C phase space. If we have that > and >, then we have a QED-lke theory as shown n Fgure 3.5. In these models the -functon s postve at least up to the perturbaton theory can not be performed anymore (.e. = g =4 & ). In such a model we can have a Landau pole, where the couplng g can go to nnty at a nte energy scale L as n QED (see Fgure.6). We have such a model f the condton s met > ) 4 T R > C A ; (3.) whch s N f > N C = n the fundamental representaton of SU (N C ). For <, > and > wth a lower number of avors N f than the QED-lke theores wth xed N C, the model can ow to an nteractng conformal xed pont of the renormalzaton group,.e. t s IR-conformal (constant at low energes). If the value of the couplng at that pont s less than one such we can perform perturbaton theory (.e. = g =4 ), then ths xed pont s called a Banks-Zaks xed pont. At the same tme the model s also an asymptotcally free theory at hgh energes as shown n Fgure 3.5. More speccally, we determne the xed pont from the -functon of the model n Eq. (3.8) up to two loops to be Page 74 of 93

77 CHAPTER 3. MINIMAL WALKING TECHNICOLOR g 3 (4) + g 5 = g 4 = 4 : (4) 4 + O g 7 = ) (3.) If we can arrange 4 to be smaller than, then we have <. From ths t follows when the couplng ows to the IR area where t s conformal, and thus the model s a weakly coupled wth the couplng g. In the fundamental representaton of SU (N C ), we have a Banks-Zaks xed pont ( < ), f the number of avors s between N C > N f > 34N C 3 3N ; (3.3) C 3 where the upper bound comes from requrement that < and the lower bound from the requrement that >. If we decrease the number of avors even more such that we have <, > and <, then we can obtan walkng theores. In these theores, lattce calculatons show that there s connement before the couplng reaches the xed pont as shown n left panel n Fgure 3.6. In ths xed pont we have that = g =4 >, where the couplng walks (conformal) between the energy scales TC and ETC as n rght panel n Fgure 3.6. Therefore, the technquarks are condensed when the couplng walks. For energes over ETC the model gets asymptotcally free and below TC the technquarks and -gluons conned. g Connement Walkng Asymptotc freedom g T C ET C Fgure 3.6: Schematc structure of the -functon (the left panel) and the gauge couplng (the rght panel) whch has a connement at low energy, a walkng phase between the energy scales TC and ETC, and asymptotc freedom at hgh energes. For even lower number of avors ( < and < ), we obtan QCD-lke theores where there s connement at low energes and asymptotc freedom at hgh energes as llustrated n Fgure 3.5. In the fundamental representaton of a non-abelan gauge theory wth gauge group SU (N C ) we have a QCD-lke theory f the number of avor s both Page 75 of 93

78 CHAPTER 3. MINIMAL WALKING TECHNICOLOR N f < N C and N f < 34N 3 C 3N C 3 ; (3.4) whch come from the nequaltes < and <, respectvely. It s dcult to show where the dstncton between walkng and QCD-lke models, where the two theores der from one another s, because the couplng s large. It requres non-perturbatve methods to determne ths dstncton. However, we can magne a walkng model as shown n Fgure 3.6. In such a model we can magne that the couplng has walked down to an energy scale whch s the same as one of the fermon mass. In that case, the number of avors s eectvely decreased wth one. If the theory s stll n the walkng regon n Fgure 3.5, then maybe the couplng wll walk agan untl t reaches the mass of the next fermon. Thus, the number of avors s agan decreased by one. In ths way, the couplng contnue untl the theory s moved down to a QCD-lke regon n Fgure 3.5, where the couplng blows up at low energes wthout walkng. Let us add an extra term to the Lagrangan n Eq. (3.7), whch s the four pont operator whch comes from an underlyng ETC theory as shown n Fgure 3. for a heavy gauge boson and n Fgure 3.3 for a heavy Hggs boson n prevous secton about ECT models. Thus, the Lagrangan s now L = Q D Q 4 Tr[G G ] ETC Q L Q R q L t R + h.c.; (3.5) where Q L;R = (U L;R D L;R ) T are the technfermons and q L = (t L ; b L ) T are the thrd generaton of SM quarks. In such a model we have for decreasng energy below the TC scale, TC, we get a condensaton of technquarks, and thus we obtan the top mass term ETC Q L Q R t L t R! hq L Q R TC t L t R m t t L t R ; (3.6) ETC where hq L Q R TC s the technquark condensate at TC scale, and the top mass renormalzed at the TC energy scale s where f s the pon decay constant. m t ( TC) = hq LQ R TC ETC 4f 3 = ; (3.7) ETC The two scales, TC and ETC, can be connected usng the renormalzaton group equaton (Eq. (3.8) n [5]) as follows hqq ETC = exp ETC d(ln )(()) TC! hqq TC ; (3.8) where s the anomalous dmenson of hqq (a scalng exponent), whch s non-perturbatve determned from the partcular techncolor model. If we have a QCD-lke asymptotcally free gauge theory, then at large energes and hqq ETC hqq TC. If we have a walkng theory as n Fgure 3.6, then the couplng s walkng from ETC down to TC. In ths case, the ne structure constant s constant n ths conformal wndow, and therefore the anomalous dmmenson s also constant,.e. that Page 76 of 93

79 CHAPTER 3. MINIMAL WALKING TECHNICOLOR exp ETC d(ln ) TC! = exp In ths case, the condensate s rescaled as follows hq L Q R ETC = ETC TC ln ETC TC ETC = : (3.9) TC hq L Q R TC ; (3.) where hq L Q R ETC s the condensate at the ETC energy scale ETC. Thus, the top mass renormalzed at ETC s m hq LQ R ETC hq LQ R TC ETC t ( ETC) = = : (3.) ETC ETC TC Thus, the rst advantage wth a walkng model s that we can lft the fermon mass by havng a large derence between the scales TC and ETC,.e. we wsh that the conformal wndows are large enough to generate the derent SM fermon masses. We can also make the mass herarchy between the fermon generaton by havng derent ETC for the derent fermons. A problem wth the ETC theores s that these theores generate four SM fermons operators, whch can be wrtten as ETC that contrbute to the avor-changng neutral currents (FCNCs), and e.g. q L q R q L q R ; (3.) to the K K oscllaton whch gves a small volaton of CP. The second advantage wth a walkng theory s that the ETC can be adjusted very hgh wthout changng the fermon masses accordng to Eq. (3.) f the anomalous dmenson s near = otherwse the masses of fermons become too small. Therefore, the FCNCs n Eq. (3.) can be suppressed by ncreasng the derent ETC. These derent ETC gve rse to the parameters n the CKM matrx Vj q n Eq. (.36). 3.5 Wenberg Sum Rules and the S Parameter The eectve model descrbed untl now has a number of free parameters whch are xed by a assocated underlyng dynamcs. In ths secton, we assume that the underlyng theory s a four dmensonal asymptotcally free gauge theory wth only fermonc elds transformng accordng to arbtrary representaton of the gauge group. The Wenberg sum rules (WSR) can be used to reduce the number of unknown parameters of such a model. The followng dscusson below s for the chral symmetry breakng pattern SU(N f )L SU(N f )R! SU(N f )V, but t can easly be generzed to any breakng pattern. To derve these sum rules we dene the tme ordered two-pont functon as the derence of vector current and axal-vector current correlaton functon a;b (q) d 4 xe q x hj;v a (x)j ;V b () hj ;A a (x)j ;A b () h Im a;b ;V Im a;b ;A ; (3.3) where a; b = ; : : : ; N f are the avor labels and the currents are Page 77 of 93

80 CHAPTER 3. MINIMAL WALKING TECHNICOLOR J a ;V = qt a q; J a ;A = qt a 5 q: (3.4) where T a are the global SU (N f ) generators. In the chral lmt (where the masses of the quarks go to zero), we have that whch obeys the unsubtracted dsperson relaton a;b (q) = (q q g q ) ab (q ); (3.5) (Q ) = ds Im (s) s + Q ; (3.6) where Q = q > (Eq. (58) n Ref. []). We assume that the underlyng theory s asymptotcally free above an energy scale, therefore the behavor of (Q ) s the same as n QCD at asymptotcally hgh momenta, we have that (Q ) Q 6 (see Ref. [39]). Thus, n Eqs. (7.6)-(7.7) n Appendx C-3, by expandng the rght-hand sde of Eq. (3.6) leads to the rst and the second Wenberg sum rule (WSR), whch are dsim (s) = ; dssim (s) = : (3.7) We break the ntegraton n the WSRs nto the regon wth low lyng resonances and the regon from ths regon up to. Ths energy scale s dened such that above ths scale asymptotc freedom sets n. The contrbuton over wll be neglgble. In the rst regon whch extends from zero to a threshold, where the ntegral s saturated by pngbs, massve vector and axal vector states. Wenberg assumed n hs orgn paper n Ref. [4] that there s only a sngle narrow resonant state wth zero wdth n the vector and axal-vector spectral functons, whch contrbute to the sum rules,.e. Im V (s) = f V (s m V ) + : : : ; Im A (s) = f A(s m A ) + f (s) + : : : ; (3.8) and totally we have Im (s) = fv (s m V ) f A(s m A ) f (s) + : : : ; (3.9) where f V, f A and f are the vector, axal mesons and the massless pon decay constant, respectvely, and m V and m A are the vector and axal-vector masses, respectvely. By nsertng the spectral functon wth nnte narrow resonances n Eq. (3.9) nto the rst WSR n Eq. (3.7), we obtan the relaton fv fa = f : (3.3) A more general relaton would replace the left hand sde of ths relaton wth a sum over all the vector and the axal-vector states. Ths WSR holds for both runnng and walkng dynamcs. In the second regon whch extends from to encodes also the conformal propertes of the theory, Page 78 of 93

81 CHAPTER 3. MINIMAL WALKING TECHNICOLOR whch s the confornal regon. The second WSR receves also mportant contrbutons from ths conformal regon. Accordng to Eq. () n Ref. [4], the second WSR gves the relaton f V m V f Am A ' a 8 d(r) f 4 ; (3.3) where a = O() whch s expected to be a postve coecent, and d(r) s the dmenson of the representaton of the underlyng fermons. As for the rst WSR, generally the left-hand sde of the second WSR wll be a sum over vector and axal states. The two WSRs can be combned, whch (see Eq. 7.3 n Appendx C-3) gves m V m A ' f a 8 m fa d(r)f V : (3.3) For example n a N f -avor model, the EW symmetry s gauged and embedded n the avor symmetry, SU(N f )L SU(N f )R = SU(N f )V SU(N f )A. When the chral symmetry breakng happens, then the avor symmetry breaks to a pure vectoral symmetry group,.e. SU(N f )LSU(N f )R! SU(N f )V. Thus, the correlaton functon n Eq. (3.3) s zero after the chral symmetry breakng. For techncolor models the EW symmetry wll break at the same energy scale as the chral symmetry breakng. Therefore, the correlaton functon s a measure for the EW symmetry breakng. Hence, by knowng the correlaton functon we can calculate the PesknTakeuch parameter called S parameter (dened n Eq. (7.34) n Appendx I), whch s an EW parameter that descrbes how much the EW symmetry s broken. In Eq. (5.) n Ref. [4] the correlaton functon s lnked to the S parameter (precson parameter). The S parameter s related to the absorptve part of the vector-vector mnus axal-axal vacuum polarzaton (VV-AA vacuum polarzaton), whch s gven by ds f S = 4 s Im (s) = 4 V m V fa m A ; (3.33) where Im s obtaned by subtractng the GB contrbutons from Im. By usng the result n Eq. (3.3) 8 S ' 4f a d(r)m V m A f : (3.34) m V + m A The last term arse from the conformal regon (from the scale up to ) s expected to be of the same order of the two other terms and negatve. Thus, t s much reduced relatve to QCD-lke theores. It s another advantage havng a walkng techncolor model, because such a model reduces the S parameter, whch s measured to be S = :5 : accordng to the LEP experments (from Eq. (.7) n Ref. [73]). The S parameters of the varous walkng techncolor models can be calculated numercally, and thus t can be tested whether these values are consstent wth the expermental data from LEP experments. The correlaton functon n Eq. (3.3) n a strong nteractng gauge theory wth the currents n Eq. (3.4) can be calculated by lattce methods from the parameters n the eectve model whch we can use to calculate the S parameter. The mass and decay constants of the vector and axal-vector partcles can also be calculated numercally, and therefore we have that the S parameter can also be determned accordng to Eq. (3.33. In Ref. [67] the vector and the axal-vector masses are calculated on lattce for the SU () gauge theory Page 79 of 93

82 CHAPTER 3. MINIMAL WALKING TECHNICOLOR wth N f = avors of fermons n the fundamental representaton. The results are m V =f TC 3:(:) and m A =f TC 4:5(3:6) (combnng statstcal and systematc errors), where the pseudoscalar decay constant s f TC = v EW = 46 GeV. Thus, the masses are m V 3: TeV and m A 3:6 TeV. In Ref. [68] these masses are also been calculated on lattce for the SU() gauge theory wth N f = avors of fermons n the adjont representaton,.e. lke the MWT model. For these models, the results for the T-B lattce are m V =f TC :38(3) and m V =m A :67(5), whch gve the correspondng masses m V 585 GeV and m A 874 GeV. These vector and axal-vector partcles are lghter than when the fermons are n the fundamental representaton. The masses can stll be above the expermental constrants, because ther couplng constants can be correspondng smaller. The decay constants, f V and f A, can not yet be calculated, because they are harder to calculate than the masses. In the future, t wll be possble to calculate the decay constants and therefore also the S parameter numercally from the parameters of the eectve model. In that way we can test the varous techncolor models by comparng these results wth the expermental result from the LEP experments. 3.6 Chapter Concluson We have provded an extenson of the Standard Model whch embodes mnmal walkng techncolor models and ther nterplay wth the partcles n the Standard Model, the fermons and the EW gauge bosons. The extenson of the Standard Model conssts of the relevant low energy eectve degrees of freedom, scalars, pseudoscalars as well as spn- partcles, whch are lnked to the underlyng mnmal walkng theory. It s called mnmal because we have the mnmal number of technfermons gauged under the EW ground (only two technfermons). The number of technfermons n turn s constraned by electroweak precson measurements, because a hgher number of technfermons contrbute correspondngly wth a hgher number of loop contrbutons to the EW parameters n Fgure 7.7 n Appendx I and thus larger EW parameters. Frstly, we have constructed an underlyng model of a techncolor model and ts eectve model wth two technfermons and techngluons both n the adjont representaton of SU()TC. Secondly, we have extended ths theory wth an extended gauge group SU(N ETC ) whch couples the SM fermons to the partcles n the techncolor model. Thrdly, because of for example the masses of the fermons are too small compared to expermental results, then we have dscussed the possblty of the dynamcs of walkng techncolor models. These models can provde the needed larger fermon masses, ther mass herarchy and the needed suppresson of the FCNCs. Fnally, the relevant EW parameter called S s been derved whch depends on the parameters of the eectve model, e.g. the decay constants and masses of the vector and axal-vector scalars. In future, ths parameter can be calculated numercally from the parameters of the eectve model, and n that way we can test the varous techncolor models. The walkng dynamcs can also reduce the S parameter, such that t t wth the expermental result from the LEP experments. Page 8 of 93

83 Chapter 4 Composte Hggs Dynamcs In ths chapter, we wll provde an uned descrpton of models of composte Hggs dynamcs, where the Hggs can be emerge ether as a massve exctaton of the condensate n techncolor models or as a pseudo- Goldstone boson n so-called composte Hggs models. Ths depends on the way the electroweak symmetry, G EW = SU()L U()Y, s embedded n the global symmetry group, G. In prevous secton and Ref. [], we had a techncolor model, where the EW symmetry s broken, SU()LU()Y! U() Q, smultaneously wth the chral symmetry breakng, SU(4)! SO(4). The classcaton of relevant underlyng gauge theory for techncolor models appeared n Ref. [49]. The contrary to these techncolor models are composte Hggs models, whch are classed n Refs. [5, 5], where the unbroken symmetry H must contan the SM electroweak group G EW. In the tradtonal techncolor setup, the Hggs boson s dented wth the lghtest scalar exctaton of the fermon condensate, e.g. the techn-. These techncolor models are not able to provde mass to the SM fermons and therefore a new sector must be added. Ths new sector can modfy the mass of the techncolor Hggs, typcally reducng t as n Ref. [48]. Another possblty s to use vacuum algnment (dscussed n Refs. [6, 7]) to algn the vacuum such that the Hggs sector does not break the EW symmetry. In ths case the Hggs boson s dented wth one of the Goldstone bosons of the chral symmetry breakng. The challenges are not only to provde mass to the SM fermons, but also to construct a Hggs potental that provdes mass to the Hggs Goldstone boson by spontaneously EW symmetry breakng. We wll mostly follow Ref. [] n ths chapter. We wll analyze models consstng of two Drac fermons whch transform accordng to the fundamental representaton of an SU() gauge group. We wll nvestgate the avor symmetry breakng pattern SO(6) = SU(4)! Sp(4) = SO(5), where the coset SU(4)=Sp(4) = SO(6)=SO(5) contans ve Goldstone bosons (GBs). The GBs decompose nto (; ) + (; ) of the subgroup SO(4) Sp(4). Ths s because a 5-dmensonal rreducble representaton of Sp(4) decomposes nto a (; ) + (; ) of the subgroup SO(4) = SU() SU() accordng to the decomposton method wth Dynkn dagrams n Appendx J. Therefore, ths chral symmetry breakng pattern allows for a Hggs doublet. In ths analyss, we wll nvestgate the mnmal scenaro of SU(4)! Sp(4) for both a mnmal 8

84 CHAPTER 4. COMPOSITE HIGGS DYNAMICS techncolor and for a composte GB Hggs scenaro by vacuum algnment. 4. The Fundamental Lagrangan In ths model we have the chral symmetry pattern SU(4)! Sp(4) wth an underlyng SU() gauge theory wth two Drac avors whch transform as fundamental representaton of the gauge group. The underlyng Lagrangan s L = 4 F a F a + U ( D m)u + D( D m)d = 4 F a F a + U D U + D D D + m QT ( )CE Q + m (QT ( )CE Q) y ; (4.) where F a s the eld strength tensor, U and D are the two fermon Drac elds whch have the bare mass m, D s the covarant dervatve, C s the charge conjugaton operator workng on Drac ndces, s the antsymmetrc tensor workng on color ndces, and the Q vector dened n Eq. (3.7). The antsymmetrc vacuum, E, n Eq. (3.3) s used, whch breaks the symmetry of the condensate from SU(4)! Sp(4). In the case where the fermon mass s zero, m =, the Lagrangan has a global SU(4) symmetry. In the case m 6=, the global SU(4) symmetry s explctly broken to Sp(4) subgroup. We have that the Q vector transforms under an nntesmal SU(4) transformaton as Q! ( + a T a )Q, where T a are the 5 generators of SU(4) wth a = ; : : : ; 5. Therefore, we have that m QT ( )CE Q! m QT ( + a T at )( )CE ( + b T b )Q + O( ) = m QT ( )CE Q + m a Q T ( )C(T at E + E T a )Q + O( ); and thus the Lagrangan transforms as (4.) L! L + m a Q T ( )C(T at E + E T a )Q + h.c. + : : : : (4.3) Thus, the only generators that obey the equatons T at E + E T a and leave the Lagrangan n Eq. (4.) nvarant are precsely the ten Sp(4) generators as shown n Eq. (7.) and Eq. (7.6) n Appendx A. Although for m = where the Lagrangan has ts full SU(4) symmetry, there wll appear a spontaneously breakng as n QCD, whch gves a nonzero vacuum expectaton value, hu R U L + D R D L 6=. It has the same structure as the terms contanng m n Lagrangan, where the dynamcal breakng would also be SU(4)! Sp(4) as shown n Eq. (3.4). Accordng to the Nambu-Goldstone theorem, we wll acheve ve GBs from the ve broken generators. Page 8 of 93

85 CHAPTER 4. COMPOSITE HIGGS DYNAMICS 4. Electroweak Vacuum Algnment We can consder the vacua where the EW sector has been embedded, such that t does not break the EW symmetry. There are two EW nequvalent valua (dscussed n Ref. [6]), whch can not be related by an SU()L transformaton, whch are A A and B A ; (4.4) whch come from the most general vacuum n Eq. (7.5) whch s derved n Appendx C-4. We have sn() = for composte Hggs models and e = for =. In ths chapter we wll use B. There s another algnment of the condensate H = E A ; (4.5) whch breaks the EW symmetry, and thus t can be used to construct techncolor models as n Refs. [5,53], where we have sn = and = n Eq. (7.5). 4.. The B Vacuum: We have accordng to Eq. (3.5) that the unbroken generators of SU(4) for the vacuum B are dened by S a B + BS at = ; (4.6) where a = ; : : : ;, because ten of the generators of SU(4) are unbroken. The sx of these form an SU() SU() subgroup of Sp(4), whch are S ;;3 A and S 4;5;6 T A ; (4.7) where we can dentfy the EW generators wth S ;;3 for SU()W and S 6 for U()Y. Thus, we can dentfy the custodal symmetry n SU() SU() group generated by the unbroken generators S ;:::;6. The remanng four are S 7;8;9 = A and S = The ve broken generators whch s assocated wth the ve GBs are X = p X 4 = 3 A ; X = p A ; and X 5 A ; X 3 = p A : A ; A ; (4.9) Page 83 of 93

86 CHAPTER 4. COMPOSITE HIGGS DYNAMICS whch satsfy the equatons (cf. the second relaton n Eq. (7.) n Appendx A), X B BX T = ; (4.) where = ; : : : ; 5. Wth the above decomposton, we can move n the quotent SU(4)=Sp(4) around the vacuum B n followng way = e X =f B; (4.) where the elds ;;3 are the GBs eaten by the massve W and Z bosons, the uctuatons around the vacuum of 4 s dented wth the Hggs (.e. h 4 = v) and 5 = s a snglet scalar. 4.. The H Vacuum Accordng to Ref. [5] the unbroken generators of SU(4) for the vacuum H are S + S 4 ; S + S 5 ; S 3 + S 6 ; S 7;9; ; X ;;3;5 ; (4.) and the broken ones can be wrtten as S S 4 ; S S 5 ; S 3 S 6 ; S 8 and X 4 : (4.3) Accordng to Ref. [5] the vev along the drecton H breaks the SO(4) SU() SU() Sp(4) to a SU()C group wth the generators S + S 4, S + S 5 and S 3 + S 6, whch s n agreement wth SM breakng pattern, where the EW symmetry s broken and we are left wth a custodal symmetry. Therefore n ths case, we have a techncolor model, where the whole EW group s not n the unbroken group H = Sp(4) A Superposton of the two Vacua: We have now analyze the two vacuum algnment lmts, the EW Hggs vacuum algnment lmt and the techncolor lmt wth the vacua B and H, respectvely. Several of the results n ths subsecton can be found n Ref. [6]. Now, we dene the vacuum of the model to be a superposton of these two vacua above = cos B + sn H; (4.4) where t s normalzed n such a way that y =, and the angle s a free parameter whch s = for EW unbroken phase and = = for a purely techncolor model. Accordng to Eq. (7.) n Appendx A, the ve broken generators can be wrtten n the vacuum as follows Y = c X s S S 4 p ; Y = c X + s S S 5 p ; Y 3 = c X 3 + s S 3 S 6 p ; Y 4 = X 4 and Y 5 = c X 5 s S 8 ; (4.5) where c = cos and s = sn. These ve broken generators satsfy the equatons Y Y T = wth = ; : : : ; 5: (4.6) Page 84 of 93

87 CHAPTER 4. COMPOSITE HIGGS DYNAMICS The ten unbroken generators are shown n Eq. (7.) n Appendx A. Here are the rst three generators Y ;;3 assocated to the GBs that become the longtudnal degrees of freedom of W and Z gauge bosons. If we work n untary gauge we use only the elds h and explctly whch s assocated to the generators Y 4;5. Thus, we can wrte = e (hy 4 +Y 5 )=f : (4.7) Here h can be dented as the Hggs boson (for sn 6= ). Some studes have as a composte dark matter canddate. The knetc eectve Lagrangan of wth nteractons to the gauge bosons va mnmal couplng s expanded n Appendx C-4. The knetc term of s gven n Eq. (7.6) as f Tr[(D ) y D ] = (@ h) + (@ ) + 48f [ ) ]+ g W + W + (g + g )Z Z f s + s f c h s 8 4f (h + ) p h + O(f 3 ); f (h + ) + (4.8) where the covarant dervatve of expressed as follows D gw a (Sa + S at ) g B (S 6 + S 6T ); (4.9) whch s derved n Appendx C-4 n Eqs. (7.63)-(7.66) such that the knetc-gauge term s nvarant under the gauge transformatons. From the expanson above we can dentfy the masses of the W and Z gauge bosons, whch are where m W = gv= n the SM, thus the vev s m W = g f s and m Z = (g + g )f s = m W =c W ; (4.) p v = fs : (4.) We can also dentfy the couplngs between the Hggs h and the gauge bosons, g hw W =gm W c = g SM hw W c ; g hzz =p g + g m Z c = g SM hzzc ; g hhw W = g c 4 g hhzz =g hhw W =c W ; = g SM hhw W c ; (4.) and couplngs between and the gauge bosons, g W W = 4 g s = gsm hhw W s g ZZ =g W W =c W : (4.3) Page 85 of 93

88 CHAPTER 4. COMPOSITE HIGGS DYNAMICS It can be noted that the knetc term of s nvarant under the Z transformaton!, and therefore wll be stable. Because s stable, then n Ref. [54] they study as a composte dark matter canddate. 4.3 Loop Induced Hggs Potental Generally, we have the avor symmetry breakng pattern G! H. In the techncolor lmt we always need that SU()W U()Y G and U()Q H. Therefore, the chral symmetry breakng happens smultaneously wth the EW symmetry breakng,.e. TC = v EW. In addton, we need that there can be found a trplet of GBs whch are absorbed as the longtudnal degrees of freedom of the weak gauge bosons n the quotent G=H. Furthermore, we must dentfy a custodal symmetry n unbroken symmetry group SU () C H. However, n the composte Hggs lmt we always need that SU()WU()Y G, SU()WU()Y H and agan a custodal symmetry n the unbroken symmetry group SU () C H. Therefore, the EW symmetry s unbroken after the chral symmetry breakng. Thus, we need a SU()W doublet (a Hggs doublet) n the quotent G=H, whch contrbutes wth the SM Hggs boson and the three GBs eaten by the weak gauge bosons. Therefore, t s needed that we nduce a Hggs potental between the energy scale of the chral symmetry breakng and the EW symmetry breakng. In ths secton we wll derve such a Hggs potental, whch s nduced by gauge one-loops, top-yukawa one-loop and an explct mass term. The dynamcs does not tell about where the condensate s algned n the SU(4) space n the above theory. As we wll see the gauge nteracton loop, the top-yukawa loop and the loop from an explct mass term wll nduce a Hggs potental. The breakng of the avor symmetry SU(4)! Sp(4) wll be communcated to the GBs va these loops, whch wll nduce a Hggs potental that determnes the value of the vacuum algnment angle n Eq. (4.4). These loop-nduced potental for ths model has also been calculated n Refs. [6, 55] Gauge Contrbutons We start to derve the contrbutons to the one-loop potental of the gauge bosons. To do ths we construct the lowest order operator whch s nvarant under the avor symmetry SU(4). To construct ths operator we need to wrte out the knetc term of n Eq. (4.8) wth ts nteractons wth gauge bosons va mnmal couplng, whch yelds f Tr f Tr h h (D ) y D = (@ ) gw a (@ ) y (S a + S at ) + gw a ( y S a + S at )@ g B (@ ) y (S 6 + S 6T ) + g B ( y S 6 + S 6T y )@ + (4.4) g W a W b ( y S a + S at y )(S b + S bt ) + gg W a B ( y S a + S at y )(S 6 + S 6T )+ g gb W a ( y S 6 + S 6T y )(S a + S at ) + g B B ( y S 6 + S 6T y )(S 6 + S 6T ) ; Page 86 of 93

89 CHAPTER 4. COMPOSITE HIGGS DYNAMICS where the covarant dervatve of s D gw a (Sa + S at ) g B (S 6 + S 6T ): (4.5) The gauge generators of SU()L are S ;;3, whle the one for U()Y s S 6. The two terms wth one W a boson cancel each other as follows Tr h gw a (@ ) y (S a + S at ) + gw a ( y S a + S at y )@ = gw a Tr = gw a Tr = gw a Tr = gw a Tr h h h h = gw a Tr = gw a Tr (@ ) y (S a + S at ) + ( y S a + S at y )@ (@ ) y S a (@ ) y S at + y S + S at S a (@ ) y S at (@ ) y + S a (@ ) y + S at S a (@ ) y S at (@ ) y + S a (@ ) y S at (@ ) y h h S a (@ ) y S at (@ ) y S a f (@ hy 4 Y 5 ) y + f (@ hy 4 Y 5 )S at y (4.6) = ; because y = y = and ( y ) = (@ ) y + (@ ) y = )= (@ ) y = (@ ) y : (4.7) The two terms wth one B boson also cancel as follows Tr h g B (@ ) y (S 6 + S 6T ) + g B ( y S 6 + S 6T y )@ = g B Tr h S 6 f (@ hy 4 Y 5 ) y + f (@ hy 4 Y 5 )S 6T y = : (4.8) Therefore, the knetc term of ncludng ts nteractons wth the gauge bosons n Eq. (4.4) can be wrtten as f Tr f Tr h h (D ) y D = (@ ) + g W a W b ( y S a + S at y )(S b + S bt )+ gg W a B ( y S a + S at y )(S 6 + S 6T ) + g gb W a ( y S 6 + S 6T y )(S a + S at )+ g B B ( y S 6 + S 6T y )(S 6 + S 6T ) whch wll be used to estmate the gauge loop contrbuton to the Hggs potental. ; (4.9) W ;;3 ; B p q q M () Gauge M () Gauge q q Fgure 4.: In left panel we have the contrbutons to the one-loop potental of the gauge boson loops, whch can be eectvely drawn as the dagram n rght panel. Page 87 of 93

90 CHAPTER 4. COMPOSITE HIGGS DYNAMICS The contrbuton to the one-loop potental of the SU() gauge boson loops as shown n Fgure 4. can be estmated from the followng term n the knetc eectve Lagrangan, whch can be wrtten as h f g W a W b Tr ( y S a + S at y )(S b + S bt ) =f g W a W b Tr h y S a S b + y S a S bt + S at y S b + S at y S bt =f g W a W b Tr h y S a S b S a S b S a S b + S at y S bt =f g W a W b Tr =f g W a W b h ab Tr hs a S b (S b ) S a (S a ) S b + S a S b (S a ) S b ; (4.3) where we have used that T =, and S at = S a because the generators are hermtan. The second term of ths expresson gves an eectve vertex wth two elds and two SU() gauge bosons elds. The Lagrangan term for ths eectve vertex can be wrtten as f g ~ C g 3X a= W a W a Tr [S a (S a ) ] ; (4.3) where the factor ~ C g s a form factor of the vertex, whch only can be determned by non-perturbatve methods, e.g. lattce methods. The external lnes of the two gauge bosons can be put together to make a loop as shown n left panel n Fgure 4., where there s ntegrated over the momemtum of the gauge boson. Thus, we wrte the one-loop potental of the SU() gauge bosons as V SU() = C g g f 4 3X a= Tr [S a (S a ) ] ; (4.3) whch s eectvely the dagram n rght panel n Fgure 4.. The factor C g s a unknown loop factor. Ths one-loop potental s expanded n powers of f up to quadratc terms n the elds h and n Eq n Appendx C-4 as follows V SU() = C g g f 4 3X Tr S (S ) = =C g g 3 f 4 c + 3 p f 3 c s h + 3 f (c h s ) + : : : : 6 (4.33) Analogously, the contrbuton to the one-loop potental of the U() gauge boson. The followng term n the knetc eectve Lagrangan n Eq. (4.9) can be rewrtten to be f g B B Tr h ( y S 6 + S 6T y )(S 6 + S 6T ) The Lagrangan term for ths eectve vertex can be wrtten as = f g B B h Tr (S 6 ) S 6 : (4.34) f g ~ C g B B Tr S 6 (S 6 ) : (4.35) By ntegratng over the momentum of the loop n left panel n Fgure 4. gves V U() = C g g f 4 Tr S 6 (S 6 ) : (4.36) Page 88 of 93

91 CHAPTER 4. COMPOSITE HIGGS DYNAMICS By expandng n powers of f we obtan n Eq. (7.77) n Appendx C-4 that V U() = C g g f 4 Tr S 6 (S 6 ) =C g g f 4 c + p f 3 c s h + (4.37) f (c h s ) + : : : 6 To determnng the unknown loop factor C g n Eq. (4.33) and Eq. (4.37) we can expand the followng trace as follows Tr S (S ) = c h + : : : ; (4.38) where = ; ; 3 for W a loops and = 6 for B loop, and c are coecents n the front of the quadratc term of the Hggs h. Thus, from Eq. (4.3) and Eq. (4.3) we have V SU() = f g ~ C g V SU() = C g g f 4 3X a= 3X a= W a W a Tr [S a (S a ) ] = f g Tr [S a (S a ) ] = C g g f 4 3X a= 3X a= c a h + : : : : W a W a c a h + : : : ; From the Feynman rules of Eq. (4.39), the ampltudes of the dagrams n Fgure 4. are (4.39) M () Gauge = ( f g ~ C g c g ) M () Gauge = C gg f 4 c ; d 4 p () 4 ( g ) p = 8f g ~ C g c 6 = ~ C g f g c ; (4.4) where the ntegral s solved n Eq. (.), and = 4f s a cuto where the condensate s meltng. Because these ampltudes are equal to each other, then we can solate the unknown loop factor M () SU() = M(), ~ f g c C SU() g = C g g f 4 c )C g = C f ~ (4f g = ) C f ~ g = 8 C ~ g : (4.4) We can nd the value of by mnmzng the eld ndependent term V SU() + @ h;= h;= = C g (3g + g )f 4 c s = : (4.4) Because the loop factor C g s postve, then ths part of the potental has a mnmum at =. Ths mnmum does not break the EW symmetry, therefore the vacuum s algned n the composte Hggs lmt. It can also be noted that the lnear term of the Hggs h s always proportonal to the dervatve of the potental, and thus ths term vanshes at the mnmum Top Contrbuton Now, we wll calculate the eects on the vacuum algnment from a top-loop contrbuton to the potental. We assume that the top mass s generated by the four-fermon operator (cf. Eq. (3.) n Ref. [6]) y t t (Qt c ) y T P + h.c.; (4.43) Page 89 of 93

92 CHAPTER 4. COMPOSITE HIGGS DYNAMICS where s an SU()L ndex, Q s an SU()L doublet, t c s the charge conjugated of the top eld, are the technfermons, t s a new dynamcal scale, and the projectors P select the components of the object T that transform as an SU()L. t t T C t t Fgure 4.: When the technfermons condense at the energy TC, then the four-fermon operator n Eq. (4.43) generates a new operator n Eq. (4.45). These projectors can be wrtten as (cf. Eq. (3.3) n Ref. [6]) P = C A ; P = C A : (4.44) When the technfermons condense at TC, ths four-fermon operator generates a new operator as shown n Fgure 4. wth two top external lnes and one external lne. Ths gves the operator yt C ~ t f (Qt c ) y Tr(P ) + h.c. yt C ~ t fs + p c h 6f s (h + ) + : : : t R t c L; (4.45) where y t s proportonal to y t (4f ) = t, and the factor ~ C t s a form factor of the vertex, whch only can be determned by non-perturbatve methods, e.g. lattce methods. The expanson of ths operator generates the top mass, m t = y tfs, from the rst term and the top-yukawa couplng from the second term when 6=, whch s where we have used that v = p fs from Eq. (4.). t = y tc p = m tc v ; (4.46) q p q t t p q M () T op M () T op q q Fgure 4.3: In left panel we have the contrbuton to the one-loop potental of the top quark loop, whch can be eectvely drawn as the dagram n rght panel. From the above operator we can construct the contrbuton of the top-loop to the Hggs potental by puttng two operators together as shown n left panel n Fgure 4.3. Accordng to Eq. (7.78) n Appendx C-4, ths contrbuton s Page 9 of 93

93 CHAPTER 4. COMPOSITE HIGGS DYNAMICS V top = C t y t f 4 = C t y t X [Tr(P )] = f 4 s + p f 3 c s h + f (c h s ) + : : : ; 8 (4.47) where there s ntegrated over the momentum n the top-loop whch gves the unknown loop factor C t. Ths loop factor can be determned by expandng the followng trace as follows Tr(P ) = c h + : : : ; (4.48) where c are coecents n the front of the lnear term of the Hggs h. Thus, we can rewrte y tf (Qt c ) y Tr(P ) + h.c. = y tf (Qt c ) y c h + + h.c.; V top = C t y t f 4 X = [Tr(P )] = C t y t f 4 X = [c h] + : : : : (4.49) From the Feynman rules of Eq. (4.49), the ampltudes of the dagrams n Fgure 4.3 are M () top =( )3(y tf ~ C t c ) d 4 p () 4 Tr[=p(=p = q)] p (p q) = 3(y tf ~ C t c ) 4 6 ~ C t = 3 4 (y tfc ) ~ C t ; M () top = C ty t f 4 (c ) ; (4.5) where the ntegral s solved n Eq. (.), and = 4f s a cuto where the condensate s smeltng. Because these ampltudes are equal to each other, thus we can solate the unknown loop factor M () top = M() top, 3 4 (y tf C ~ t c ) = C t yt f 4 (c ) )C t = 3 4 f ~ C t = ~ C t : (4.5) As for the gauge loops, we can nd the value of by mnmzng the eld ndependent term of Eq. (4.47) as = C t yt f 4 s c = : h;= The loop factor C t s postve lke the loop factor C g. Thus, the mnmum s located at = =, whch breaks the EW symmetry at TC scale where the technfermons condense. We have the top-loop contrbuton to the Hggs potental such that t prefers the vacuum n the drecton whch corresponds to the TC vacuum lmt. In the TC vacuum lmt ( = =), the pngb h can not be a Hggs-lke partcle, because the lnear couplngs of h to the gauge bosons and to the top vansh accordng to Eq. (4.) and Eq. (4.46). Therefore, the physcal Hggs state can not be one of the pngbs, and t must be the lghtest composte scalar state. The two pngb h and can nstead be lnkng together nto a complex d-technquark GB, whch can be wrtten as h +. We have also n the TC vacuum lmt that the masses of h and are degenerate, and accordng to Eq. (4.33), Eq. (4.37) and Eq. (4.47) ther loop-nduced mass s Page 9 of 93

94 CHAPTER 4. COMPOSITE HIGGS DYNAMICS m DM = m h = m = f 4 3g C t yt + g C g : (4.53) Thus, the weak gauge nteractons msalgn the TC vacuum, where the h and are massve and the three pngbs ;;3 are massless. Whle the top-loop correctons realgn the vacuum n the TC drecton, where we have the massve h + eld and the three massless pngbs ;;3. Therefore, the top-loop correctons provde a postve mass to a potental dark matter canddate h +. The complex state, h +, s a good dark matter canddate, because t has a U() symmetry n the knetc Lagrangan n Eq. (4.8), and then t s stable. Ths state has been used extensvely for dark matter model buldng n Refs. [53, 5658]. As we have just seen, the pngb h can not be used as the Hggs boson n the TC-lmt. As we conclude n Eq. (.68), t gves rse to a problem, because we can not produce a composte partcle wth the mass of the Hggs boson, m h = 5 GeV, unless the number of techncolor s very hgh. Ths n turn s constraned by electroweak precson measurements, because a hgher number of techncolor contrbute correspondngly wth a hgher number of loop contrbutons to the EW parameters n Fgure 7.7 n Appendx I and thus larger EW parameters. Therefore, we want to algn the vacuum away from the TC-lmt. Ths can be done by another possble contrbuton comng from an explct term that break the SU(4) avor symmetry whch can gve a mass splt between h and. For example, a mass term of the technfermons that explctly breaks the SU(4) avor symmetry Explct Breakng of SU(4) Mass terms for the technfermons wth a gauge nvarant masses can be ad-hoc added to gve mass to. These are an another sources to the Hggs potental that break explctly SU(4). Such sources wll algn the vacuum away from the TC-lmt. A mass term whch s algned wth the condensate B wth the mass M = B (cf. Eq. (3.7) n Ref. [6]) can be wrtten as T B ; (4.54) whch gves the contrbuton to the Hggs potental, whch accordng to Eq. (7.78) s p V m =C m f 4 Tr( B ) = C m 4f 4 c + f 3 s h + f c (h + ) + : : : ; (4.55) 4 where the coecent C m can have both sgns. Ths potental term contrbutes to push the vacuum away from the TC-lmt ( = =). We want to mnmze the eld-ndependent potental terms of the total Hggs potental n Eq. (4.33), Eq. (4.37), Eq. (4.47) and Eq. (4.55), whch s 3g + g V () = C g c + y t C t s + 4C mc f 4 3g + g = C g c + y t C t ( c ) + 4C mc f 4 : (4.56) Page 9 of 93

95 CHAPTER 4. COMPOSITE HIGGS DYNAMICS We dene that X t yt C 3g + g t C g ; X m C m : (4.57) Thus, the eld-ndependent potental terms can be rewrtten to be whch s mnmzed for V () = X t c 4X m c + constant; (4.58) = ; c = X m X t : (4.59) We can dentfy the loop-nduced masses of the pngbs h and n the Hggs potental n Eq. (4.33), Eq. (4.37), Eq. (4.47) and Eq. (4.55), whch are m h = 3g + g = f 8 f c C g 4 f c C t y t 4 X t( c ) + X mc ; m = 3g + g 8 f s C g + 4 f s C t y t + f c C m + f c C m f = X t( c 4 ) + X mc : Wth the soluton =, where the EW symmetry s unbroken, the masses read (4.6) (4.6) Ths soluton s stable f X t < X m. m h =f For the soluton c = X m =X t, the masses read m h =f 4 4 (X m X t ); m = f X m: (4.6) X t 4X m X t = f 4 s X t ; m = f 4 X t: (4.63) For m h > we need that X t > jx m j (.e. c < ) whch corresponds to broken EW symmetry. We recover the relaton m = m h =s as n Ref. [6]. The pngb Hggs mass can be rewrtten to m h =f 4 s X f t = 4 s yt C 3g + g t C f m t g = 4 s f s C tm t = m W + m Z C g C tm t 4 4m = t C t 4 = C tm t 4 :79 C g C t ; C m W + m Z t C 4f s g (8:385 GeV) + (9:88 GeV) C g 4(7:44 GeV) C t (4.64) where we have used that the top mass m t = y tfs and Eq. (4.). Ths shows that the contrbuton from the gauge loops s typcally smaller than the top-loop, f we are assumng that C g C t. If we neglect the contrbuton from the gauge loops, then we have m h C tm t 4 ) C t 4m h m t 4(5:9 GeV) = : (4.65) (7:44 GeV) Therefore, we have that the loop factor n the top-loop contrbuton s C t. The Hggs couplngs to Page 93 of 93

96 CHAPTER 4. COMPOSITE HIGGS DYNAMICS the gauge bosons n Eq. (4.) are now well constraned by LHC data (see Eq. (.44)). Thus, a realstc value of must be small,.e. c ) X t X m : (4.66) Thus, the vacuum s algned close to the composte Hggs lmt (.e. = ). In ths case, we can produce small enough mass of the Hggs boson, and large enough mass gap up to the next lghtest resonance (the pngb ) wth the mass m = m h =s (cf. Eq. (4.63)) because s s small. Wth ths vacuum algnment there s no more a dark matter canddate, because the pngb s not stable anymore. Ths s, because the added potental has no Z symmetry of for hgher expanson order of the potental, and therefore can decay. However, another possble problem arses, because the vacuum algnment angle s s needed to be ne-tuned. 4.4 Fne-Tunng of the Model The denton n Eq. (.6) of the quantty for how much a observable O s ne-tuned compared to a parameter s O < max; (4.67) where for example we can choose the maxmal tolerance for ne-tunng to be max =. We can calculate ths quantty for the ne-tunng of the top-couplng, y t, to the vacuum algnment angle, s, n the above model. We solate s n Eq. (4.59), whch gves c = X m X t, s 4X = m : (4.68) Xt Ths gves the ne-tunng quantty s BG;y t y t y = t t 8X m X 3 t = y t s 8Xm C t : (4.69) Xt 3 By nsertng C t and X t X m = C m (cf. Eq. (4.65) and Eq. (4.66) respectvely) nto ths ne-tunng quantty, we obtan s BG;y t y t ; (4.7) C m s where the coecent n the front of =s s roughly at the order of unty. Thus, the ne-tunng quantty s very large, because s s very small cf. Eq. (4.66). Therefore, we need to ne-tune the parameter s very much. Accordng to Eq. (4.66) we need to ne-tune between the top-loop and the explct breakng of SU(4) contrbutons. Ths ne-tunng must be nduced by another completely derent mechansm. 4.5 Chapter Concluson In ths chapter, we have constructed a model of a composte Hggs based on a strongly nteractng gauge theory wth fermonc matter elds, where we have studed smultaneously models of pngb Hggses and Page 94 of 93

97 CHAPTER 4. COMPOSITE HIGGS DYNAMICS TC models. In the TC lmt the Hggs s dented wth the lghtest scalar resonance, techn-, of the dynamcs, whle away from the TC lmt t s dented wth one of the pngbs, h. We have focused on the example of the avor symmetry pattern, SU(4)! Sp(4). We have studed the most mnmal strongly coupled gauge theory, the SU() gauge theory, wth two technfermons transformng as a fundamental representaton of the gauge group. The coset SU (4)=Sp(4) contans ve pngbs, where three of them are eaten by the massve weak gauge bosons. The fate of the remanng pngbs depends on the algnment of the vacuum. In the TC algnment, they form a complex dark matter canddate, h +. In the algnment away from the TC-lmt one of the two pngbs plays the role as Hggs boson, whle the another pngb can not play the role as dark matter state, because t s not expected to be stable. Our analyss shows that the algnment n the TC-lmt s more natural, because there s small or no ne-tunng between the contrbuton from the top-loop and the explct breakng term and thus the vacuum algnment angle s. However, n TC-lmt there s the problem that the Hggs boson must be the lghtest scalar resonance, techn-, whch seems to have too much mass to be dented wth the Hggs boson. What we have won wth ths knd of models are that we have got a dynamcal explanaton of the scalar nature of the Hggs boson and smultaneously we have removed the problem wth a too heavy Hggs canddate as n the TC models. We obtan these advantages at expense of a new ne-tunng problem of the vacuum algnment and that we need to add a Hggs potental ad hoc. Addtonally, we have stll not completely removed the EW herarchy problem, because we need one or more scalars to produce the fermon masses as n the TC models. Page 95 of 93

98 Chapter 5 Partally Composte Hggs Dynamcs We wll n ths chapter examne the EW symmetry breakng based on the mxture of a fundamental Hggs doublet, H, and an composte pseudo-nambu Goldstone doublet. The condensaton of the strongly nteractng fermons trggers a vev for the fundamental Hggs doublet so that the EW symmetry breakng stll arses dynamcally but the EW scale and the Hggs partcle arses as a mxture of composte and fundamental sectors. Ths dea s due to 't Hooft n Ref. [] and were orgnally termed bosonc Techncolor (BTC) models (mentoned later n Ref. [7]). Bosonc because of the fundamental Hggs boson doublet and techncolor because of the composte doublet. However, just as above where we studed CH models as a msalgnment of a TC model, also here we wll study such a msalgnment wth a fundamental scalar present, as s done n Ref. [3] and term n partcally composte Hggs (PCH). In ths chapter the possble trvalty and the vacuum stablty of ths model wll be nvestgated by calculatng the runnng of the fundamental Hggs self-couplng, h. From the runnng we can determne the energy scale, where we have a Landau pole or an unstable vacuum. In that way we can nvestgate n what part of the parameter space the model s self-consstent. 5. The Fundamental Lagrangan In the Ref. [3], the authors consder a mnmal non-supersymmetrc BTC model wth a sngle fundamental Hggs doublet, H, and realgn the vacuum nto a PCH model va an electroweak preservng mass term. The mnmal TC sector contans technfermons transformng as fundamental representatons under the gauge group SU()TC, where the left- and rght-handed technfermons transform as doublets and snglets under SU()W, respectvely. Ther gauge quantum numbers are shown n Table 5.. Thus, the technfermons transform n the fundamental representaton of the techncolor gauge group SU ()TC. When the weak nteracton s turned o, then the model has a global SU(4) under whch the fourcomponent object Q L = (U L D ~ L U ~ L D L ) T (5.) transforms n the fundamental representatons explaned before where we construct the four-component 96

99 CHAPTER 5. PARTIALLY COMPOSITE HIGGS DYNAMICS vector n Eq. (3.7). The condensate of the technfermons can be wrtten as hq a LQ T;b L C / ab ; (5.) where acts on the TC ndces, C = dag( ; ; ; ) acts on the left-handed Weyl spnors n Q L, and the most general Sp(4) vacuum s gven by e cos sn sn e cos A ; (5.3) whch s derved n Appendx C-4. The angle [; ], s a phase whch volates CP, and breaks SU(4) spontaneously to ts subgroup Sp(4), as explaned n Appendx C-4. For sn = the condensate s purely SU()L breakng (the techncolor lmt), whle for sn = the electroweak symmetry s unbroken (the composte-goldstone lmt). SU()TC SU()W U()Y (U L ; D L ) T ~U L = ~D L += Table 5.: Technfermon gauge quantum numbers. The knetc terms for the SM fermons, technfermons Q L ncludng the electroweak and TC nteractons, and all gauge elds are wrtten as follows L K = D + Q y L ( A G a a = 4 )Q L 4 F a F a (5.4) where = (; ~ ) and the covarant dervatve s gven by D g Y B g a W a g 3 a The elektroweak (EW) gauge bosons are ncluded n the matrx A g W a a g B 3 Ga : (5.5) A ; (5.6) and F a A A a + gf abc A b A c whch s the eld strength tensor of one of the gauge elds, where F abc = for the photon eld, F abc = abc for the W, Z and the techngluon elds, and F abc = f abc for the gluon elds. The new n ths PCH model compared to the TC and CH models n the prevous two chapters s a Hggs sector wth a fundamental Hggs doublet, H. The knetc Lagrangan for ths Hggs doublet and ts potental n SU(4) notaton s gven by Page 97 of 93

100 CHAPTER 5. PARTIALLY COMPOSITE HIGGS DYNAMICS L H = Tr[(D H) y D H] V (H) = Tr[(D H) y D H] + Q T LC (M + )Q L + h.c. m HjHj h jhj 4 : (5.7) The fundamental Hggs doublet H has a postve mass parameter m H and the quartc self-couplng, h, and can be wrtten (same form as the Hggs doublet n the SM n Eq. (.8)) as H(x) = h (x) + h (x) v + h (x) 3 (x) A ; (5.8) where the elds h (~ h ) are the scalar components of H wth the vacuum expectaton value (vev) v jhhj. The gauge covarant dervatve reads D g Y B g a W a : (5.9) The matrces M and are two 4 4 matrces, whch contan the gauge-snglet masses m ; Hggs-Yukawa couplngs to the two technfermons U;D, and can be wrtten as follows where M m m A ; H H T H = U ( h + v h 3 ) D( h + h ) U (h + h ) D( h + v + h 3 ) and the A ; (5.) A : (5.) The Yukawa terms wth the technfermons come from the second term wth the technfermons n Eq. (5.7). These terms gve rse to the technfermon masses, whch are m U = U v = p and m D = D v= p. The mass terms can be wrtten as m U L D L + m ~ U L ~ D L + m U U L ~ U L + m D D L ~ D L ; (5.) where the terms wth the masses m and m are terms whch can be wrtten because they are gauge nvarant. To explan these terms we need a new scalar, call t S. Ths must be an EW snglet, and hence we can not use the scalar H. Therefore, we need to construct terms lke SU L D L and S ~ U L ~ D L, where the masses are m = v S = p and m = v S = p. The last part of the total Lagrangan s the Yukawa Lagrangan terms of the fundamental Hggs H to the SM fermons. The Yukawa Lagrangan can be wrtten accordng to Eq. (.73) n terms of Weyl spnors as follows L Y = u " j q Lj H u R d q LH d R e l L H e R + h.c.; (5.3) where u, d and e are the Yukawa couplngs, and q L = (u L ; d L ) T and l L = ( L ; e L ) T are the left-handed weak doublets of the quarks and the leptons, respectvely, and u R, d R and e R are the rght-handed weak snglets of the up-type quarks, the down-type quarks and the electron-type leptons respectvely. Fnally, we can collect all Lagrangan terms above n the total fundamental Lagrangan of ths theory, Page 98 of 93

101 CHAPTER 5. PARTIALLY COMPOSITE HIGGS DYNAMICS whch s L = D + Q y L (@ A G a a = 4 )Q L 4 F a F a + Tr[(D H) y D H] + Q T LC (M + )Q L + h.c. m HjHj h jhj 4 u " j q Lj H u R d q y L H d R e l L H e R + h.c. : (5.4) 5. Constructon of the Eectve Lagrangan Intally, we have the global symmetry group G = SU(4), whch s spontaneously broken to the global group H = Sp(4). The SU(4)=Sp(4) coset contans ve broken generators, X a, and ten unbroken generators, S, whch satsfy the relatons X a X at = and S + S T = cf. Eq. (7.) n Appendx A. The broken and the unbroken generators, X a and S, are lsted as T? a and T k n Eq. (7.) and Eq. (7.) n Appendx A, respectvely. The 5-dmensonal rreducble representaton of Sp(4) can be decomposed nto a (; ) + (; ) under the subgroup SO(4) = SU() SU() of Sp(4) = SO(5) as shown n Appendx J by usng Dynkn dagrams. The SU() ; can be dented wth the generators (S a S a+3 )= p wth a = ; ; 3. These generators are reduced to SU()L;R generators n Eq. (3.34) n the sn! lmt. Therefore, S a wth a = ; ; 3 form the sospn group SU()V = SU() + = SU()L+R. The exponental realzaton of ths 5-dmensonal representaton of Sp(4) = SO(5) of the Nambu Goldstone bosons a can be wrtten as p = exp( a X a =f ); (5.5) where parameter f s the TC decay constant n the chral lmt. The object parametrzes n the coset G=H, whle the exponent of parametrzes n the algebra. Ths object transforms as! UV y (U; ) accordng to Eq. (7.59) n Appendx K, where the global transformatons are U SU(4) and V Sp(4). We use the realzaton whch transforms as! UV y nstead of the non-lnear representaton = T (the same as n Eq. (4.7) n Chapter 4) whch transforms as! UU T, because transforms both wth U and V, and therefore the symmetres are more explct, and we wsh to know how the objects transform under V Sp(4). Therefore, these buldng blocks can be used as blocks wthout thnkng about that further. We have a freedom to choose between these two representatons, = exp( p a X a =f ) and = T, because of the powerful eld theoretc theorem called Haag's theorem (page n Ref. [4]). It states that there s a representaton ndependence, f two elds are related non-lnearly, e.g. ' = F () wth F () =, as the above representatons, then the same expermental observables result. Here we have small uctuatons of the elds ' and,.e. we can expand F () = F () + : : : : = If ths s the case, you can wrte all eectve Lagrangan terms n terms of the specc representaton whch respect the same global symmetres and gauge symmetres, and then you wll obtan the same Page 99 of 93

102 CHAPTER 5. PARTIALLY COMPOSITE HIGGS DYNAMICS physcs from these terms. E.g. for scalar elds, the mass terms n terms of the eld ' can be rewrtten n terms of the other representaton to rst order n F () as follows m ' = m F () = m (F () + : : : ); (5.7) and also for the knetc terms (@ ') = ((@ )F () F ()) = (@ ) + : : : : (5.8) Ths s also shown for the nteracton terms n Ref. [69]. By usng ths representaton, the knetc Lagrangan terms are expressed n terms of the quantty called the Maurer Cartan -form (Eq. (7.6) n Appendx K) C = y D ; (5.9) whch lves n the algebra. The sem-covarant dervatve s D A ; (5.) wth the gauge elds A n Eq. (5.6). The quantty C transforms lke a Sp(4) gauge eld as C! V (C )V y (see n Appendx K). We can project C onto elds parallel and perpendcular to the unbroken Sp(4) drecton as (cf. Eq. (7.6) n Appendx K) C? = Tr(C X a )X a C k = Tr(C S )S ; (5.) whch are a 5-plet and -plet of Sp(4), respectvely, and C = C? +Ck. These transform homogeneously and lke a gauge elds accordng to Eq. (7.65) and Eq. (7.66),.e. C?! V C? V y ; C k! V (C k )V y : (5.) Furthermore, we can dene the quantty = T (M + ) h.c.; (5.3) whch transforms under Sp(4) as! V V T (Eq. (7.8) n Appendx C-5). Wth these buldng blocks n hand we can construct a Sp(4) nvarant Lagrangan. The leadng O(p ) chral Lagrangan s L f () = Tr(C? C? ) + 4f 3 Z Tr( + ); (5.4) where Z :47 accordng to a N c = N f = lattce study n Ref. [3]. The rst term n the TC eectve Lagrangan n Eq. (5.4) and the Hggs knetc term n Eq. (5.7) yeld the EW scale (v EW = 46 GeV) Page of 93

103 CHAPTER 5. PARTIALLY COMPOSITE HIGGS DYNAMICS v EW = f sn + v ; (5.5) whch s derved n the Eqs. (7.8)-(7.89) n Appendx C-5. The angle s the same angle as n the vacuum matrx n Eq. (5.3), whch tells about whch drecton the vacuum s algned. 5.3 The Vacuum Algnment We wll mnmze the O(p ) potental V () e = 8f 3 Z hm cos UD v sn = p + m H v + h 4 v4 ; (5.6) whch s derved n Eqs. (7.9)-(7.3) n Appendx C-5 from second TC Lagrangan term n Eq. (5.4) and the fundamental Hggs potental n Eq. (5.7). We have dened that m m +m, UD U + D and m UD m U + m D = v( U + D )= p v UD = p. The mnmzng of the potental () =8f 3 Z h From the rst vacuum condton we obtan p m sn UD v cos = = ; (5.7) = 8f 3 Z UD sn = p + m Hv + h v 3 = : (5.8) tan = m UD m : (5.9) The m U;D mass terms tend to algn the vacuum n the drecton of the TC vacuum lmt ( = =) as the top-loop potental n Eq. (4.47). On the other hand, the m ; mass terms prefer the drecton of the EW-unbroken vacuum lmt ( = ). They correspond to EW preservng mass operators, as opposed to the Drac mass terms m U;D as seen n Eq. (5.), smlarly to the explct mass term that break the SU(4) symmetry n Eq. (4.55). From the second vacuum condton n Eq. (5.7), we obtan an expresson for the Hggs self-couplng h = 4p Z f 3 sn m H v v 3 : (5.3) In Table 5.3 the mportant expressons above and ther orgns are collected. Expresson The Orgn of the Expresson tan = m UD m Vacuum algnment 4 p f 3 Z UD sn + m H v + hv 3 = Vacuum algnment tan v f sn v EW = f sn + v Denton The TC and Hggs gauge-knetc terms Table 5.: Important expressons and ther orgns n ths partally composte Hggs model. Page of 93

104 CHAPTER 5. PARTIALLY COMPOSITE HIGGS DYNAMICS 5.4 Scalar Resonances In ths secton, we determne the masses of the varous scalar resonances. In Eqs. (7.3)-(7.39) n Appendx C-5 the mass matrces of the scalars are calculated explctly from the TC eectve Lagrangan term 4f 3 Z Tr( + ) and from the terms of the fundamental Hggs potental m HjHj h jhj 4. In Eqs. (7.3)-(7.334) the mass egenstates and ther masses are derved by dagonalzng the mass matrces below. Accordng to Eq. (7.33), the charged scalar mass matrx n the bass ( + h ; + ) s M + m H + hv m H t h v t m H t h v t m H t + hv t where h = ( h h )=p and = ( )= p and tan t A = (m H + h v t t t A ; (5.3) v f sn : (5.3) The mass egenstates of the charged scalar mass matrx n Eq. (5.3) are the two charged pon states wth the masses (Eq. 7.33) G = s h + c and ~ = c h + s ; (5.33) m G = and m ~ = (m H + hv )=c : (5.34) Accordng to Eq. (7.38), the neutral scalar mass matrx n the bass ( 3 h ; 3 ) s M 3 m H + hv m H t h v t m H t h v t m H t + hv t A = (m H + h v t t t A : (5.35) The mass egenstates of the other neutral scalar mass matrx n Eq. (5.35) have the same form as the two charges pon states, whch are G 3 = s 3 h + c 3 and ~ 3 = c 3 h + s 3 (5.36) wth the masses m G 3 = and m ~ 3 = (m H + hv )=c : (5.37) The mass of the 5 whch does not mx wth the other scalars s accordng to Eq. (7.39) m 5 = t (m H + hv ); (5.38) Fnally, accordng to Eq. (7.39) we have that the neutral scalar mass matrx n the bass ( h ; 4 ) s gven by M h = m c t c t t A + h c t c t t A ; (5.39) Page of 93

105 CHAPTER 5. PARTIALLY COMPOSITE HIGGS DYNAMICS The mass egenstates of the neutral Hggs mass matrx n Eq. (5.39) are h = c h s 4 and h = s h + c 4 : (5.4) Accordng to Eq. (7.35) n Appendx C-5, the mxng angle between the two components of h ; s where " 3 h v =m H. For " = ( h = ) we obtan t c t ( + "=3) = (5.4) + " t ( + "=3): tan = cos tan tan = cos tan ; (5.4) whch s the expresson below Eq. (4) n Ref. [3]. The masses of h ; (cf. Eq. (7.3)) are m h ;h = m H =c + "( + t =3) r =c + "( + t =3) 4 c t ( + "=3) + t ( + ")( + "=3) : (5.43) The h and h mass states are the lght and heavy neural Hggs, respectvely. The h s the canddate to the Hggs n the SM, whch s a lnear combnaton of the fundamental Hggs h and the composte pngb component 4. For small " (3 h v m H ) and small s we obtan from Eq. (5.43) (see n Eq. (7.37) m h = m H s s + " 3 s ( c ) + O(" ): (5.44) For h = (" = ) we obtan that m h = m H sn sn ; (5.45) n accordance wth Eq. (6) n Ref. [3] n the lmt s c. The mass of the Hggs state h depends on sn, and therefore t acqures ts mass from a strong sector vacuum msalgnment lke n composte pngb Hggs models. From Eq. (5.43) we can solate the self-couplng, whch gves h =a(b + c); (5.46) where the coecents are a ==(t v 4 (c 6)); b = v (3m h 4m Ht + c m Ht + m h t ); (5.47) c = v qm 4 h (9 6t + t4 ) + m H m h (8c t t ) + 4(c m4 h t + m4 H t4 + m H m h t 4 ); whch s renormalzed at the SM Hggs mass m h,.e. h = h (m h ). The mass egenstates of the scalars above and ther masses n Eqs. (5.38)-(5.37) are collected n Table 5.3. The mass egenstates G and G 3 are the NGBs that become absorbed as the longtudnal degrees of freedom of the weak gauge bosons. Page 3 of 93

106 CHAPTER 5. PARTIALLY COMPOSITE HIGGS DYNAMICS The Mass Egenstate The Mass h ; = c h s 4 Equaton (5.43) ~ = c h + s m ~ = (m H + hv )=c G = s h + c m G = ~ 3 = c h 3 + s 3 m ~ 3 = (m H + hv )=c G 3 = s h 3 + c 3 m G 3 = 5 m 5 = t (m H + hv ) Table 5.3: The mass egenstates and ther masses n ths partally composte Hggs model. 5.5 The Normalzaton Factors The couplng of the lght Hggs (h ) to the weak gauge bosons (V = W ; Z) h V V and the Yukawa couplng to the lght Hggs (h ) h ff are normalzed to the SM ones as follows where g PCH V V = g v=, g SM V V g PCH V V h + c s fg 4 W + W = g v EW =, PCH f = V g SM hv V h W + W ; (5.48) PCH f h ff = F SM f h ff; (5.49) = m f p =v and SM f = m f p =vew. Frst and second term n Eq. (5.48) come from the gauge-knetc Lagrangan for the fundamental Hggs n Eq. (5.7) and the rst term n Eq. (5.4), respectvely. We can see that both h and 4 couple to the weak gauge bosons, whle t s only h that couples to the fermons. The normalzaton factors are (Eq. (7.348)) V = c s s c c ; F = c =s ; (5.5) whch are derved n Eqs. (7.335)-(7.348) n Appendx C The Angles n the Model The sgns of sne, cosne and tangent of both the angle, and and the reasons to these sgns are shown n Table 5.4 For example, sn > and cos >, because we assume that the top-yukawa couplngs are postve. Frstly, we can assume that sn, f the top-yukawa couplng n the model s nearly the same as n the SM accordng to Eq. (7.343) n Appendx C-5. Secondly, we can also assume that sn, because the composte part 4 n the mass egenstate of the SM Hggs h n Eq. (5.4) s most domnatng. Thus, tan >. The rest of the sgns are explaned n the table above. Page 4 of 93

107 CHAPTER 5. PARTIALLY COMPOSITE HIGGS DYNAMICS Sgn of Angle Reason sn > Postve Yukawa Couplngs: h t tan > tan = v=(f sn ) = s = cos > sn ; tan > sn > sn = (f=v) tan > tan < tan = m UD =m < cos < sn >, tan < q = SM t =s > s > cos > Postve Yukawa Couplngs: PCH t = (cos = sn ) SM t > tan > sn ) tan < ) tan = cos tan > and sn sn > cos >, tan > Table 5.4: The sgns of sne, cosne and tangent of the angles, and and the reasons. 5.7 The Parameter Space Now, we wll nvestgate the parameter space of ths model. Ths s done by usng Matlab, whch can calculate the parameters above from the three nput parameters: the mass of the fundamental Hggs m H, the angle s and the angle t. Some examples of vacuum algnment are shown n Table 5.5, where the values of the derent parameters are calculated from the nput parameters m H = m p H, s and t. In these calculatons t s used that m t = 7:44 GeV, m h = 5:9 GeV and s (m Z ) = :84 from Table.3. Therefore, g 3 (m t ) = :75 and t (m t ) = :939 accordng to the rght panel of Fgure.6 and Eq. (.9), respectvely. In Table 5.5, we consder two derent masses of the fundamental Hggs, m H = GeV and m H = 3 GeV. For each of these masses we have three derent values of the angle s (.3,.5 and.5 for m H = GeV and.45,.3,.5 for m H = 3 GeV) and three derent values of the angle t (3.8,.7 and.7 for both masses m H and each value of s ). Our choce of the values of t are random and have no specal meanng. The fundamental Hggs self-couplng at the mass of the SM Hggs mass h (m h ) decreases, when the angle t ncreases for xed values of m H and s. Thus, there s an upper bound of t, where the self-couplng h (m h ) becomes negatve, and therefore the vacuum s unstable. Ths upper bound of t ncreases wth decreasng s. For example, for the xed values, m H = GeV and t = 3:8, the Hggs self-couplng h (m h ) s negatve for s = :5 and postve for s = :5. In addton, ths upper bound of t also ncreases wth decreasng m H. For example, for the xed values, s = :5 and t = 3:8, the Hggs self-couplng h (m h ) s negatve for m H = GeV and postve for m H = 3 GeV. Page 5 of 93

108 CHAPTER 5. PARTIALLY COMPOSITE HIGGS DYNAMICS E m H s t s f m UD v m ~ m h m 5 h (m h ) log GeV :3 3:8 : : :634 :3 :7 : :8 : :96 :3 :7 : :3 : :66 :5 3:8 : : :48 :5 :7 :68 7: : :4 :747 :5 :7 : :4 : :586 :5 3:8 : :5 : :33 6:57 :5 :7 : :3 : :495 6:738 :5 :7 : :5 : :9 3 :45 3:8 : : : :9 3 :45 :7 : : : :446 :99 3 :45 :7 : :9 : :7 3 :3 3:8 :9 46 : : :78 3:7 3 :3 :7 : :3 : :395 4:74 3 :3 :7 : : : :75 3 :5 3:8 : :6 : :36 6:368 3 :5 :7 :68 :8 : :4469 7:97 3 :5 :7 : : : :3 Table 5.5: Examples of vacuum algnment, scalar spectrum, and the vacuum nstablty energes E (all masses are n GeV). The parameters are calculated by Matlab. In these calculatons t s used that m t = 7:44 GeV, m h = 5:9 GeV and s (m Z ) = :84 from Table.3. Therefore, g 3 (m t ) = :75 and t (m t ) = :939 accordng to the rght panel of Fgure.6 and Eq. (.9), respectvely. The masses of the pngbs are expermentally constraned, for example the mass of 5 whch s a pure composte partcle. The mass of 5 decreases both wth decreasng m H and s, respectvely. For example for xed, s = :3 and t = :7, the mass of 5 s 77 GeV for m H = GeV and 39 GeV for m H = 3 GeV. In ths example f the mass s 39 GeV for m H = 3 GeV, then the pngb 5 mght have been observed dependng on the magntudes of ts couplngs to the other partcles. Another mportant observaton s that ether s or t should be decreased to make the theory stable (.e. where h (m h ) s postve), such that the mass m H can be pushed up. Thus, the herarchy problem n ths model s smaller compared to the SM, because the mass of the new fundamental scalar, H, s smaller ne-tuned compared to the SM Hggs mass due to ts larger mass. The allowed parameter space wll be reduced even further n the next secton, where we nvestgate the vacuum stablty of ths model. Page 6 of 93

109 CHAPTER 5. PARTIALLY COMPOSITE HIGGS DYNAMICS 5.8 The Vacuum Stablty The parameter space nvestgated n prevous secton can be constraned even further by studyng the energy where the model becomes unstable,.e. the energy, when the Hggs self-couplng h s runnng to negatve values at a vacuum nstablty energy E. The evoluton of the self-couplng, h, s descrbed by ts beta functon, whch s derved to rst order n Appendx E. It s gven n Eq. (7.95). In the followng calculatons we wll couple the fundamental Hggs to the SM gauge bosons, such that the beta functon of the self-couplng s (cf. Eq. 3.5 n Ref. [8]) = (4) 4 h 6 4 t + h t 3 h g 9 h g g4 + (g + g ) ; (5.5) where the -functon of the top-yukawa couplng, t, to rst-loop order ncludng the couplngs to the SM gauge bosons s (cf. Eq. 3.3 n Ref. [8]) = (4) 9 3 t 7 g g + 8g 3 The -functon of the SM gauge couplngs to rst-order are (cf. Eq. 3. n Ref. [8]) t : = 4 96 g3 = 9 96 g3 ; = 7 6 g3 3: (5.53) The top-yukawa couplng n the model whch s moded wth the factor = sn compared to the SM accordng to Eq. (7.343),.e. the top-yukawa couplng s PCH t = SM t sn : (5.54) Therefore, the top-yukawa couplng n ths model s equal to or larger than n the SM. In the -functon of h the term proportonal to 4 t wll pull t down to negatve values. Thus, wth a larger top-yukawa couplng, the self-couplng wll be negatve at lower energy than n the SM,.e. a lower nstablty energy E compared to the SM (E SM 8 GeV, cf. Fgure.8). These -functons n Eqs. (5.5)-(5.53) are solved for Hggs self-couplng h as coupled derental equatons usng Euler's method by Matlab. Matlab s used to calculatng the vacuum nstablty energy that also calculates the derent parameters by usng the equatons above from the three nput parameters m H, s and t. The value of the self-couplng h (m h ) s calculated at the SM Hggs mass m h by usng Eq. (5.46), but for smplcty ts runnng s started at the mass of the top quark m t = 7:44 GeV because all the other couplngs are renormalzed at ths energy. Ths s a good approxmaton because the runnng between the mass m h = 5:9 GeV and m t = 7:44 s small. The top couplng t (m t ) s calculated wth Eq. (.9) by nsertng the central values n Table.3 for m t = 7:44 GeV, m h = 5:9 GeV and s (m Z ) = :84, whch gves t (m t ) = :939. The vacuum nstablty energes of the vacuum algnment examples n Table 5.5 are calculated n Matlab. The values are shown n the table. It can be observed that ponts n the parameter space can be very unstable as expected. For example, the vacuum algnment wth m H = GeV, s = :5 and t = :7 s already unstable at the energes over E = 56 GeV though ths vacuum algnment gves a Page 7 of 93

110 CHAPTER 5. PARTIALLY COMPOSITE HIGGS DYNAMICS postve self-couplng h = :4. Ths s very much below the vacuum nstablty energy n the SM. Ths can agan be allevated by decreasng ether m H, s or t even more such that the vacuum nstablty s ncrease as observed n Table 5.5. For example, for xed values, m H = GeV and t = :7, we have log (E =GeV) = :747 for s = :5 and log (E =GeV) = 6:738 for s = :5. Therefore, the parameter space s lmted even further by consderng the runnng of the self-couplng. m H = 5 GeV m H = 3 GeV log (E=GeV) m H = GeV log (E=GeV) log (E=GeV) log (E=GeV) m H = 5 GeV m H = 4 GeV m H = 5 GeV log (E=GeV) log (E=GeV) Fgure 5.: Color plots of the vacuum nstablty energes E as functon of sn and tan for the masses of the fundamental Hggs m H = 5; 3; ; 5; 4 and 5 GeV calculated and plotted n Matlab. The black plot s where the decay constant s f = v EW. In these calculatons t s used that m t = 7:44 GeV, m h = 5:9 GeV and s (m Z ) = :84 from Table.3. Therefore g 3 (m t ) = :75 and t (m t ) = :939 accordng to the rght panel of Fgure.6 and Eq. (.9), respectvely. Page 8 of 93

111 CHAPTER 5. PARTIALLY COMPOSITE HIGGS DYNAMICS In Fgure 5. color plots of the vacuum nstablty energes E are plotted as functon of s and t for varous masses m H. In these color plots t can be observed that the stable parameter space s gettng smaller when the mass m H ncreases. Furthermore, the model s most stable n the regon where the angles s and t are small. The blue areas n the plots represent models where the self-couplng ether s already negatve at EW scale (.e. the model s unstable) or too large to perturbatve calculatons (.e. h = h =4 > ). In the last case, we can not say anythng about how t s runnng, t can gve a Landau pole, be unstable or be stable all the way up to the Planck scale, because the perturbaton theory breaks down. To say somethng about these ponts we need non-perturbatve methods for example lattce methods. m H = 3 GeV m H = 3 GeV log (h) h Fgure 5.: Color plots of the self-couplng h (m h ) renormalzed at the mass of SM Hggs as functon of sn and tan for the mass m H = 3 GeV. Calculated and plotted by Matlab. Left panel: the angle t goes from. to. Rght panel: the angle t goes from to 5. In Fgure 5. the self-couplng h (m h ) for m H = 3 GeV s plotted for : < t < n left panel and < t < 5 n rght panel, respectvely. In upper left panel n Fgure 5.3 the plot of the nstablty energes for m H = 3 GeV s plotted. The ponts n the blue regon for t <, the self-couplngs h (m h ) are too large to perturbaton theory for ponts under around t = :8 as shown n left panel n Fgure 5.. The ponts n the blue regon for t > have all self-couplngs whch are negatve as shown n rght panel n Fgure 5., and therefore these models are unstable. Futhermore, there s a narrow strp n the plot for m H = 5 GeV n Fgure 5., where the self-couplngs are postve and small enough to use perturbaton theory all the way up to the Planck scale. m H [GeV] s max (E & E SM ) :5 : :5 : :5 :5 Table 5.6: The values of the s angle under whch the theory can be nearly equally or more stable than the SM n our one-loop approxmaton (.e. E & E SM 8 GeV) for varous masses m H. The values are read o from Fgure 5., and therefore these values are approxmately values. Page 9 of 93

112 CHAPTER 5. PARTIALLY COMPOSITE HIGGS DYNAMICS If we want a model whch s just as good or better than the SM from the vacuum stablty pont of vew, then the nstablty energy must be equal to or larger than E SM 8 GeV to rst-loop order as shown n Fgure.8. It can be read o n the color plots n Fgure 5. under whch maxmum s angle for all t we can have an nstablty energy of the theory whch s nearly equal or larger than the value n SM. These s angles are shown n Table 5.6 for varous m H. For the mass m H = 3 GeV we must have s. : to have a theory whch s at least as stable as the SM. However, for m H = 5 GeV the s angle s needed to be below one percent (=) and for m H = 5 GeV below one permlle (=). Ths gves maybe a new ne-tunng problem of s, when the mass m H s adjusted up to reduce the the EW herarchy problem n SM (v EW =M Planck 7 ). In that way the queston why the Hggs boson s so much lghter than the Planck mass (or the grand uncaton energy or a heavy neutrno mass scale) s closer to beng answered, where the reducton of ths herarchy problem s traded to smallness of the angle s. m H = 3 GeV log (E=GeV) m H = 3 GeV V > :85 F > :75 log (E=GeV) m H = 3 GeV m 5 > 5 GeV m H = 3 GeV m 5 > GeV log (E=GeV) log (E=GeV) Fgure 5.3: Color plots of the vacuum nstablty energes E as functon of sn and tan for the masses of the fundamental Hggs m H = 3 GeV wth varous expermental constrants (no constrants, V > :85 and > :75, m 5 > 5 GeV, and m 5 > GeV respectvely). The black plot s where the pon decay constant s f = v EW. Calculated and plotted by Matlab. In these calculatons t s used that m t = 7:44 GeV, m h = 5:9 GeV and s (m Z ) = :84 from Table.3. Therefore g 3 (m t ) = :75 and t (m t ) = :939 accordng to the rght panel of Fgure.6 and Eq. (.9), respectvely. Page of 93

113 CHAPTER 5. PARTIALLY COMPOSITE HIGGS DYNAMICS In upper left panel n Fgure 5.3 the nstablty energes for m H = 3 GeV are shown wthout expermental constrants. In upper rght panel the expermental constrants are added for V and F n Eq. (5.5) whch from the LHC are at the level of 5% and 5% respectvely Ref. [35],.e. V > :85 and F > :75. It removes some of the ponts at low t, but t does not aect the results much. These constrants have also been ncluded n all the plots n Fgure 5.. In the two lower panels the constrants m 5 > 5 GeV and m 5 > GeV are added for the mass of the pngb 5, respectvely. For these constrants some of the stable ponts n the parameter space are removed, but t has only a small eect of the vablty of the model. Ths constrant has smaller and smaller eect for larger masses m H accordng to the values n Table 5.5. Therefore, these constrants have nearly no consequences. The black curves n both Fgure 5. and Fgure 5.3 are where the pon decay constant s f = v EW, whch gves s as functon of t from the two lowest expressons n table 5.3 as follows v EW = f s + v = f s + f t s ) f = f = v EW ) s = q : + t v EW ( + t )s ; (5.55) Below these curves we have the constrant f > v EW, whch we want to achevng large enough masses to the resonances. Ths constrant restrcts not the parameter space more, because the most stable models are below the curves n Fgure Chapter Concluson We conclude that the parameter space s ne-tuned n order to obtan a vacuum stable model wth a large mass m H especally the s angle. The motvaton was to reduce or completely remove the electroweak herarchy problem n SM. Ths can only be done at expense of a new ne-tunng problem. However, a soluton to ths ne-tunng problem could be that the fundamental Hggs mass parameter m H s protected by ntroducng supersymmetry (SUSY), such that we can have a low m H protected by SUSY and thus no ne-tuned s. Page of 93

114 Chapter 6 Conclusons We began ths thess by consderng the potental problems n the SM. Frstly, we can conclude that the observed Hggs partcle at least partally cures volaton of untarty n the weak sector and the custodal symmetry s mnmal broken by the Yukawa sector n the SM. The parameters descrbng devaton of the observed boson's couplng to the W bosons and the breakng of the custodal symmetry are measured to be W = :9 +: : and = :6 :9 at the LHC and LEP experments, respectvely. They are normalzed such that they are both one n the SM. If W was one then the observed boson would be fully responsble for untarzng W L W L scatterng as shown n Eq. (.45). Accordng to expermental data from LEP experments, the EW precson parameters are measured to be S = :5 : and T = :8 : (the parameter s related to T ), whch are normalzed to be zero n the SM. Therefore, the SM are consstent wth the measurements of these parameters. We have shown perturbatvely that although n solaton the SM Hggs sector s trval ths s moded when the top-yukawa couplng s ncluded. Instead, the SM s possbly vacuum unstable, because the Hggs self-couplng,, becomes negatve at energes above ts nstablty energy, E 8 GeV, as computed n the one-loop approxmaton. Despte of all the successes of the SM t cannot explan all observatons and there are a varous reasons that t cannot be the ultmate theory of Nature. Ths ncludes the exstence of neutrno masses, baryogeness, dark matter and dark energy. An mportant reason to beleve the SM s not a complete descrpton of EW symmetry breakng s that the Hggs boson s unnatural, because ts bare mass must be very ne-tuned to an absurd precson of about : 3 (accordng to the BG quantty n Eq. (.7)) to acheve the correct physcal mass. Ths naturalness problem s addressed n composte formulatons of the Hggs mechansm. By ntroducng techncolor the Hggs mechansm has a natural dynamcal orgn and s smultaneously non-trval n analogy wth chral symmetry breakng n QCD. It does not explan the orgn of SM fermon masses, and therefore we ntroduced extended techncolor to explan these masses. Such ETC models cause ther own set of problems. It s challengng to generate enough mass to the heavest fermons (n some realzatons t s already problematc to produce the mass of the charm quark), ETC contrbutes to the avor changng neutral currents (FCNC) and contrbutes to dscrepances wth precson electroweak measure-

115 CHAPTER 6. CONCLUSIONS ments. We dscussed that these potental ssues can be allevated by assumng that the gauge couplng constant of TC evolves slowly between the scales TC and ETC. These knd of models are called walkng TC models. However, the potental problem wth these TC models s that the Hggs boson s needed to be dented wth the resonance techn-, whch seems to be too heavy to play the role as the Hggs boson although see e.g. Ref. [7] for a dscusson of how top-quark correctons may change ths. Ths problem s allevated n CH models, where the vacuum s algned away from the TC vacuum. In these models we dented the Hggs boson as one of the pngbs, h, wth the desred small mass. The next lghtest resonance, pngb, has also large enough mass to explan why we have not observed t yet. Unfortunately, these models have another ssue, whch s that the top-loop and the explct mass term contrbutons to the Hggs potental seems to be ne-tuned compared to each other. Thus, we nvestgated the possblty wth a PCH model, where the Hggs boson s partally composte and fundamental. By a novel analyss of the vacuum stablty of ths model, we conclude that the parameter space s needed to be ne-tuned to obtan a vacuum stable model wth a large fundamental mass parameter m H. We had the motvaton to reduce or completely remove the EW herarchy problem n the Hggs sector, but ths can only be done at expense of a possbly new ne-tunng problem. However, ths model may be saved by ntroducng supersymmetry, such that we can have a low mass parameter m H protected by supersymmetry and thus no ne-tuned s. Page 3 of 93

116 Chapter 7 Appendces Appendx A: SU(4) generators In ths appendx we have wrtten the generator matrces of the SU(4) group. It s convenent to use the followng representaton of SU(4) S a A B B y A T A ; a = ; : : : ; 6 and X C D y D C T A ; = ; : : : ; 9 (7.) where A s hermtan, B = B T, C s hermtan and traceless and D = D T. The rst four matrces are S a = p where a = ; : : : ; 3 are the Paul matrces, and 4 = I. The next two matrces are A ; = S a = Ba B ay A ; a = ; : : : ; 4; (7.) A and 3 A ; (7.3) A ; a = 5; 6; (7.4) where B 5 = and B 6 =. The matrces S a are the generators of the SO(4) group. The last nne matrces are X = p X T D D y A ; = ; ; 3; (7.5) A ; = 4; : : : ; 9; (7.6) 4

117 CHAPTER 7. APPENDICES where D 4 = I, D 5 = I, D 6 = 3, D 7 = 3, D 8 = and D 9 =. The ten generator matrces of symplectc group Sp(4) are S a wth a = ; : : : ; 4 and X wth = 4; : : : ; 9. These generators satsfy the followng commutaton relatons (Eq. (.) n Ref. [6]) [S a ; S b ] = f abc S c [X a ; S b ] = f abc X c (7.7) [X a ; X b ] = f abc S c From the second commutaton relaton, we have that [X a ; S b ] = f abc X c ) X a S b S b X a = f abc X c ) E X a E E S b E E S b E E X a E = f abc E X c E ; (7.8) where E are the vacuum matrces for SO(4) and Sp(4), respectvely, gven n Eq. (3.3), and f t s true that S at E + E S a = ) S at = E S a E ; (7.9) X at E E X a = ) X at = E X a E ; then E X a E E S b E E S b E E X a E = f abc E X c E ) X at S bt + S bt X at = f abc X ct ) (7.) S b X a + X a S b = f abc X c ) [X a ; S b ] = f abc X c : Therefore, we have that S at E + E S a = ; X at E E X a = : (7.) Rotaton of the Sp(4) Generators Matrces nto a General Vacuum In the followng the Sp(4) and SU(4)=Sp(4) generators wll be wrtten wth a general Sp(4) metrc e cos sn sn e cos A ; (7.) whch s the general vacuum (derved n Appendx C-4). We can wrte an arbtrary SU (N ) generator as T = T k + T?, whch satsfy the relatons T k + T T k = ; T? T T? = ; (7.3) where T k and T? are the projecton of the SU(N ) generators T on parallel wth the Sp(4) generators and perpendcular to Sp(4) generators (.e. the SU(N )=Sp(N ) generators), respectvely. These are projectons n the sense that (T k )? = ; (T k ) k = T k : (7.4) Page 5 of 93

118 CHAPTER 7. APPENDICES If these projectons satsfy these condton, then we have T k + T T k = ) T k + T T? y = ; T? T T? = ) T? T T? y = (7.5) together wth (T k )? = and (T k ) k = T k gve the projecton generators T k = (T T T y); T? = (T + T T y): (7.6) To derve the form of the generators we wll start wth the EW vacuum n the composte lmt A ; (7.7) and by performng a SU(4) rotaton we obtan the general vacuum = U U T, where cos sn sn cos A SU(4): (7.8) By performng the rotaton of the generators T = U T U y, we obtan the generators n a general vacuum, because T k; + T T k; = ) U T k; U y U U T + U U T U T T k; U T = ) T k + T T k = ; (7.9) where = U U T and T k = U T k; U y, and n the same way wth the broken generators T? = U T?; U y. Thus by performng the rotaton of the generators T = U T U y, we obtan the broken generators n a general vacuum are gven by T? = p T? 4 = s c c 3 s A ; T 5? = A ; T? = p and the unbroken generators are gven by Tk = p Tk 5 = p Tk 8 c s s s A ; T k = c c s p s c s s A ; T 6 k = A ; T 9 k = p p c s A ; T 3? = s 3 c c s 3 A ; A ; (7.) A ; T 3 k = c 3 s s c A ; T 7 k = A ; T k = p p A ; T 4 k A c s 3 s 3 c A ; (7.) A ; Page 6 of 93

119 CHAPTER 7. APPENDICES Appendx B: Gauge Anomaly Cancellaton In ths secton, we wll show that the Standard Model wth a gravtatonal nteracton and the Mnmal Walkng Techncolor model are gauge anomaly free. In a gauge theory n whch gauge bosons couple to a chral current, the trangle dagrams appear n the one-loop correctons to the three-gauge-boson vertex functon (see Fgure 7.. These anomalous terms volate the Ward dentty for ths ampltude. The theores can be gauge nvarant only f these anomalous contrbutons dsappear. p ; p ; p 3 ; 5 k p 3 ; 5 k p ; p ; Fgure 7.: Feynman dagrams for the trangle anomaly The smplest Green functon where the anomales occur s the three-pont functon of two vector and one axal-vector currents, T jk (x; y; z) = ht j (x)jj (y)jk (z); (7.) where ; j; and k can take the values V; A; and P, whch requre to replace the j by the vector current j a = T a, the axal-vector current j 5a = 5 T a, and the energy-momentum four-vector p a, respectvely. The correspondng Ward denttes for a local chral transformaton = e (x) 5 (x) = e (x) 5 (7.3) can be calculated (shown n Eq. (.7.6) n Ref. [3]), whch yelds the x T V V z T V V A (x; y; z) y T V V A (x; y; z) = V V P (x; y; z) = mt (x; y; z): (7.4) Ths s what should happen, when there would be no anomales. Now, we wll calculate the trangle-graph anomaly n Fgure 7.. The two dagrams for T V V A UV-dvergent. Therefore, they must be regularzed. Ths can be done by Paul-Vllars regularzaton by subtractng the same dagrams wth mass M m. By usng the Feynman rules we get are T V V A (p d 4 k ; p ; p 3 = p p ) = 3 () 4 Tr (=k m) (=k p = m) 5 (=k + =p m) + Tr (=k m) (=k p = m) 5 (=k + =p m) (m! M ) : (7.5) Now, the ntegral s nte. In order to test the the rst two Ward denttes n Eq. (7.4) we multply Page 7 of 93

120 CHAPTER 7. APPENDICES wth p or p, and decompose =p = (=k + =p m) (=k m) = (=k = p m) + (=k m), whch gves p T V V A (p d 4 k ; p ; p 3 = p p ) = () 4 Tr (=k m) (=k p = m) 5 + (=k + =p m) (=k p = m) 5 + Tr (=k + =p m) (=k m) 5 (=k + =p m) (=k p = m) 5 (m! M ) : (7.6) By performng the shfts k! k + p and k! k + p p, we nd that the ntegrand vanshes. The same can be shown by multplyng the ntegral wth p. Consequently, the two vector Ward denttes are fullled. The axal-vector case can be studed by multplyng the three-pont functon n Eq. (7.5) wth p 3 and decompose =p 3 5 = (=p + =p ) 5 = (=k = p m) (=k + =p m) + m 5 = (=k = p m) (=k + =p m) + m 5. We get that d 4 k p T V V A 3 (p ; p ; p 3 = p p ) = () 4 mtr (=k m) (=k p = m) 5 (=k + =p m) + mtr (=k m) (=k p = m) 5 (=k + =p m) MTr (=k M ) (=k p = M ) 5 (=k + =p M ) MTr (=k M ) (=k = p M ) 5 (=k + =p M ) = (7.7) mt V V P + A ; where by replacng 5 by 5 n the dagrams n Fgure 7. we get T V V P d 4 k = Tr (=k m) (=k p = m) () 4 5 (=k + =p m) + Tr (=k m) (=k p = m) 5 (=k + =p m) ; (7.8) and the ntegral (calculated on page 74 n Ref. [3]) A = lm M! M (; )! (; ) d 4 k () 4 Tr[ (=k + M ) (=k = p + M ) 5 (=k + =p + M )] (k M )[(k p ) M ][(k + p ) M ] + = lm M! 6M " p p 6 M = " p p : (7.9) Ths ntegral s UV-nte but non-vanshng. Therefore, we have an anomaly. The moded (anomalous) Ward denttes for the regularzed one-loop vertex functon are p T V V A p 3 T V V A = p T V V A = + mt V V P " p p : (7.3) Thus the vector currents are anomaly-free whereas the axal-vector current has an anomaly from quantum uctuatons. In the case of non-abelan currents, the couplng matrces T a enter. Generally, the trangle- Page 8 of 93

121 CHAPTER 7. APPENDICES graph anomaly of the axal-vector current s Tr[fT a ; T b gt c = ] " p p A abc Therefore, the anomalous term of a trangle dagram of three gauge bosons s proportonal to (7.3) Tr[ 5 T a ft b ; T c g] = Tr[TLfT a L; b TLg] c Tr[TRfT a R; b TRg]; c (7.3) where the trace s over all fermon speces. The factor 5 s assocated wth chral currents. Ths factor s equal to for left-handed fermons and + for rght-handed fermons. The antcommutator comes from that we take the sum of the two derent trangle dagrams n whch the fermons crcle n opposte drectons n Fgure 7.. Now, we wll show that the Standard Model (wth symmetry group SU(3) SU() U()) and a model of the gravtatonal force s anomaly free. We can omt the dagrams wth three SU(3) bosons or of one SU(3) and two gravtons because all of the couplngs are left-rght symmetrc. The full set of dagrams s shown n Fgure 7.. U() SU() SU() SU(3) SU(3) U() U() U() U() U() U() SU(3) U() SU() SU() SU(3) U() SU(3) SU() Grav. U() SU() SU() SU() U() SU(3) SU() SU() SU(3) Grav. Fgure 7.: Possble gauge anomales of the SM and a model of a gravton. All of these anomales must vansh for the theory to be consstent. We have that the anomaly of three SU() gauge bosons always vanshes because of the property of Paul matrces f a ; b g = ab. There we have that Tr[ a f b ; c g] = ab Tr[ a ] = ; (7.33) and therefore the anomaly vanshes. The anomales where the dagram s contanng one SU() or SU(3) gauge boson wll always vansh, because they are proportonal to Tr[ a ] = or Tr[ a ] = where a and a are Paul and Gell-Mann matrces respectvely. The remanng nontrval anomaly dagrams are shown n Fgure 7.3. The anomaly n the upper left panel wth three U() gauge bosons s proportonal to Tr Y 3 = 3 h ( ) 3 = ; (7.34) Page 9 of 93

122 CHAPTER 7. APPENDICES where the sum nvolvng both left- and rght-handed quarks and leptons wth an extra - for the lefthanded partcles. The factor 3 counts the three color states of the quarks. U() SU() U() U() U() SU(3) SU() Grav. U() U() SU(3) Grav. Fgure 7.3: The remanng nontrval gauge anomales. The anomaly n the upper rght panel wth two SU() bosons and one U() boson s proportonal to X Tr[ a b Y ] = ab Y fl = ab 3 6 = ; (7.35) f L where the sum runs over the left-handed fermons. The anomaly n the lower left panel wth two SU(3) bosons and one U() boson s proportonal to Tr[ a b Y ] = ab X q Y q = ab = ; (7.36) where the sum s over the left-handed and rght-handed quarks. The anomaly wth two gravtons and one U() gauge boson s proportonal to Tr[Y ] = ( ) = : (7.37) Therefore the Glashow-Wenberg-Salam theory s completely free of axal vector anomales among the gauge currents. An extended gauge symmetry group of the SM model G = SU(3)C SU()TC SU()L U()Y wth the techncolor symmetry group SU()TC. Ths theory s called the Mnmal Walkng Techncolor theory. The gauge anomales cancel wth the followng generc hypercharge assgnment Y (Q y y + L ) = ; Y (U R; D R ) = ; y ; Y (L y 3y + L ) = 3 ; Y (N 3y R; E R ) = ; ; (7.38) Page of 93

123 CHAPTER 7. APPENDICES where the parameter y can take any real values. We recover the SM hypercharges for y = =3, and the electrc charge s Q = T 3 + Y, where T 3 s the weak sospn generator. We must check that the theory s gauge anomaly free. The anomaly wth three U() gauge bosons as n Eq. (7.34) s proportonal to Tr Y 3 h =3 y 3 + y+ 3 y y 3 + 3y+ 3 3y 3 + = 8 (y + y + + y + y y + y y + y + 9y 3y + + 8y 6y 9y 3y 8y + 6y) = ; (7.39) where the hypercharges n Eq. (7.38) have been used. Ths can be done for the anomaly wth two SU() bosons and one U() boson as n Eq. (7.35) whch s proportonal to Tr a b Y = ab X f L Y fl = ab 3 y 3 y = : (7.4) For the anomaly wth two SU(3) bosons and one U() boson as n Eq. (7.36) s proportonal to Tr a b Y = ab X q Y q = ab 3 y + y+ + y = ; (7.4) and for the anomaly wth two gravtons and one U() boson as n Eq. (7.37) s proportonal to Tr [Y ] = 3 y + y+ + y 3 y + 3y+ + 3y = : (7.4) Therefore the gauge anomales s cancelled n the Mnmal Walkng Techncolor theory wth the hypercharge assgnment n Eq. (7.38). Page of 93

124 CHAPTER 7. APPENDICES Appendx C: Group Representatons We have been gven the structure coecents f abc of a nonabelan group. The representaton of that group R s speced by a set of D(R) D(R) traceless hermtan matrces TR a that the commutaton relatons [TR; a TR b ] = f abc TR c (7.43) where D(R) s the dmenson of the representaton and ths commutaton relatons are the same as the orgnal generators matrces T a. These orgnal generator matrces T a 's corresponds to the fundamental representaton. If we have a untary transformaton, V T V = (T ) ; (7.44) then for V = I such that T = (T ) for every, we have that the representaton R s real. If V 6= I, we have that the representaton R s pseudoreal. If such untary matrx does not exst, the representaton R s complex. In ths case, the complex conjugate representaton R s speced by T a R = (T R a ) (7.45) It can be shown that the matrx V s only unque up to a constant. We have from Eq. (7.44) that V T V = T and QT Q = T ) V T V = QT Q ) T V = V QT Q ) V QT = T V Q ) [V Q; T ] = ) (7.46) V Q = ) Q = V From lne two to lne three n Eq. (7.46) we have used that V Q must be a constant of the unt matrx accordng to Schur's rst lemma. Therefore we have that the matrx V s only unque up to a constant. It can also be shown that the generator matrces have egenvalue + for f the representaton R s real and - for f t s pseudoreal. We have accordng to Eq. (7.44) that V T V = T ) T T V = V T (7.47) We can use Eq. (7.44) and Schur's rst lemma to get that V T V = T ) V T T = T V ) T V T = V T T T ) T V T V = V T T T V ) T V T V = V T V T ) T V T V V T V T = ) [T; V T V ] = ) V T V = a ) (7.48) V T = av ) V = av T = a V ) a = ; where a = and a = are the egenvalues for a real and pseudoreal representaton respectvely. If least Page of 93

125 CHAPTER 7. APPENDICES one generator matrx TR a (or a real lnear combnaton of them) has egenvalues that s not a = then the representaton s complex. Another mportant representaton of the compact nonabelan group s the adjont representaton A whch s gven by (T a A )bc = f abc (7.49) The structure constants f abc are real and therefore the generator matrces satsfy the condton T a A = (T a A ). Thus the adjont representaton s real. We have that the complex conjugate transforms as follows U!gU = e a T a RU ) U! U A g y AB = U A (g ) T AB = g BA U A = e a T a R U = e a ( T a R ) U = e a T a R U: (7.5) Thus, we have that U!gU = e a T a RU and (7.5) U!g U = e a T a RU; (7.5) where the generators n the complex conjugate representaton are T R = T R. From the algebra of the representaton of the group R we obtan that [T a R; T b R ] = f abc T c R ) [ T a R ; T b R ] = f abc T c R ) (7.53) [T a R ; T b R ] = f abc T c R ; whch denes that complex conjugate representatons have the same algebra,.e. that the generators T a R full the same commutaton relatons. Page 3 of 93

126 CHAPTER 7. APPENDICES Appendx D: Goldstone Theorem In ths secton we wll determne the consequence by breakng symmetres at the quantum level. The consequence s descrbed by the Goldstone theorem. The Goldstone theorem states the followng: If a symmetry group G of sze dm G s broken, then there exsts as many massless partcles as there are generators. If the group s only broken partly than only as many massless partcles appear as generators are broken. To determne the Goldstone theorem at quantum level, t s useful to nvestgate the normalzed partton functon T [J ] = Z[J ] Z[] = Z() D exp d 4 x(l + J ) ; (7.54) where L s the Lagrangan of the theory, J are the sources of the elds. The varaton of the partton functon s = Z[J ] = D exp d 4 x(l + J ) d j + S + d 4 xj ; (7.55) whch s vanshng, because the Lagrangan and the measure are nvarant under a symmetry transformaton. The rst term s the varaton of the measure whch s nvarant, and therefore t vanshes. The second term s the varaton of the acton, whch also vanshes. The thrd term must also vansh, therefore we have that d 4 xj Tk a T [J ] = ; (7.56) J k where t has been used that Z[] s constant, and T [J ] J = Z[] D exp d 4 x(l + J ) : (7.57) The relaton between the generatng functonal of Green functons T [J ] and generatng functonal of connected Green functons T c [J ] s T e Tc ) T = (e Tc ) = e Tc T c (7.58) By nsertng Eq. (7.58) nto Eq. (7.56) we get Eq. (7.56) n terms of the generatng functonal for connected Green functons e Tc d 4 xj Tk a T c [J ] = : (7.59) J k Ths can be transformed nto an equaton n terms of the generatng functonal of vertex functons. Ths s related to connected one by a Legendre transformaton [] = d 4 xj + T c [J]; h =hj [J ]j = T c[j] J ; J = [] : (7.6) Page 4 of 93

127 CHAPTER 7. APPENDICES By exchangng the dervatve and the source we get that d 4 x T kh a k = : (7.6) When the elds developng a vacuum expectaton value v, t then holds v =h = T c J [] J = [v ] By derentatng Eq. (7.6) wth respect to the eld we get the equaton d 4 x (x) T kh a k + T j (y) (x a y) (7.6) ; (7.63) where the last term vanshes snce the generators are traceless or because = j =v = J =. If we use the nverse propagator of the elds (x) j (y) [v ] = (D ) k (x y); (7.64) and that the rst term n Eq. momentum, whch yeldng (7.63) s just the Fourer-transform of the nverse propagator at zero (G ) j (p = )T a kv k = : (7.65) Thus, there must vansh as many nverse propagators as there are non-zero v. The nverse propagator at tree-level s (G ) j = j (p + m ); (7.66) whch mples that the pole mass must vansh, and therefore the propagator becomes a propagator of a massless partcle. Ths conrmes the Goldstone theorem: If you have a symmetry group G whch breaks to the subgroup H (called the stablty group of G),.e. that the coset space G=H of sze dm G=H s broken, because the elds get vacuum expectaton values v. Thus, there wll exst as many massless partcles as there are broken generators,.e. dm G=H generators. Page 5 of 93

128 CHAPTER 7. APPENDICES Appendx E: Beta Functons In ths secton, we wll calculate the -functon for the Hggs four-pont self-couplng. To calculatng the -functon, we need the Feynman rules of these Lagrangan terms 4 ( y ) t ab Q La bt R t ba Q La bt R = 6 ( (v + h) ) t p t L (v + h + 3)t R t L (v + h 3 )t R + : : : = 6 h4 + 8 ( + + t 3 p )h h(t Lt R + t Rt L ) + : : : (7.67) = 6 h4 + 8 ( + + t 3 p )h htt + : : : ; where the complex Hggs doublet s wrtten (x) = (x) + (x) v + h(x) 3 (x) A : (7.68) To determne the -functon of a couplng constant, we need to look at the relaton between the bare and renormalzed couplng of both Hggs four-pont couplng and the Yukawa couplng to the top quark, whch are = Z Z ~ ; t = Z = Z Z t ~ = t ; (7.69) whch s renormalzed usng MS scheme wth = 4e ~. The constants Z, Z, Z and Z t are the scalar wave functon, the Hggs four-pont couplng, spnor wave functon and top-yukawa couplng renormalzaton constant, respectvely. We take the logarthm of these relatons on both sdes, whch gves where we have dened ln = ln t = X X n= n= ln(z Z ) L n ( t ; ) n G n ( t ; ) n X n= ln(z = Z Z t ) + ln + ln ~; + ln t + ln ~; L n ( t ; ) n ; X n= G n ( t ; ) n : (7.7) (7.7) Thereafter, we derentate the two equaton n Eq. (7.7) wth respect to ln and multply wth and t on both sdes, respectvely, whch gves us X t = X n= n= t d t d ln d t d ln n d d ln d d ln n + n + d d ln + ; d t d ln + t: (7.7) In a renormalzable theory, we have that d=d ln and d t =d ln must be nte when!. Therefore, Page 6 of 93

129 CHAPTER 7. APPENDICES we can wrte the two equatons above as follows d t d ln = t + t ( t ; ); d d ln = + ( t ; ): (7.73) By nsertng the two expresson nto the two equatons n Eq. (7.7) we nd that the -functon of the two couplngs are ( t ; ) t ( t ; ) = L ; G ; (7.74) where the coecents to hgher orders of = must vansh. These expressons can be used to calculate the -functons, whch accordng to Eq. (7.73) encodes how the couplngs develop when the energy scale changes. We wll start to calculate the ampltudes of one-loop correcton to the Hggs four-pont vertex wth a scalar loop as shown n Fgure 7.4. The scalar n the loop s ether the Hggs (h) or one of the three would-be Goldstone bosons ( ;;3 ). There s such an one-loop correcton n both s-, t- and u-channel, whch each gves dentcal contrbuton to the -functon. In addton, the dagrams have also a symmetry factor of =, whch we multply on the ampltude. q + p p p p s-channel t-channel u-channel Fgure 7.4: The three dagrams for the one-loop correcton to the Hggs four-pont vertex wth a scalar loop, where there s ether Hggs tself or one of the three would-be Goldstone bosons n the loop. The ampltude of the these dagrams s calculated as follows M 4;scalar loop = 3 4! = 9 = 9 = 9 d 4 q () 4 q m + (q + p) m + dx dx d 4 q () 4 (x (q m + ) + x ((q + p) m + )) (x + x ) dx d 4 q () 4 (q m + + x (p + qp)) d dx 4 l () 4 (l ) = 9 ( d dx ) (4) d= () d d=4! (7.75) Page 7 of 93

130 CHAPTER 7. APPENDICES 9 (4) 9 = (4) dx dx (4) + nte = 9 log + log(4) + O() log(m x( x)p ) + log(4) + O() ; where we have used dmensonal regularzaton to regularze the loop ntegral, and we have dened m x( x)p l q + xp ) l = q + xpq + x p : (7.76) The next dagram, we wll calculate to the one-loop correcton to the Hggs four-pont vertex, s the dagram n Fgure 7.5 wth a top-loop and four external Hggs lnes. There are ve other permutatons of ths dagram, whch are obtaned by permutng the external momenta. In addton, the ampltude of these dagrams must be multpled by 3 because of the color charge of the top quark n the loop. We get also a factor, because we have a fermon loop. p p 3 q q 4 q q 3 p p 4 Fgure 7.5: One of sx dagrams wth a top-loop and four external Hggs lnes. The other ve are obtaned by permutng the external momenta. The ampltude of these dagrams s d 4 q M 4;top loop = 6 ( ) 3( t p ) 4 () 4 Tr[(=q + m)(=q p = + m)(=q p = p = + m)(=q p = + m)] (7.77) 3 (q m + )((q p ) m + )((q p p ) m + )((q p 3 ) m ; + ) where we have used the four-momentum conservaton n the vertces to derve the four-momenta n the loop, whch are p q + q 4 = ; p q 4 + q 3 = ; p 3 + q q = ; and p 4 + q q 3 = ) q = q; q = q p 3 ; q 3 = q p p ; and q 4 = q p : (7.78) We can rewrte the denomnator by ntroducng Feynman parameters as follows (q + )((q p ) + )((q p p ) + )((q p 3 ) + ) = dx dx dx 3 dx 4 6 x (q + ) + x ((q p ) + ) + x 3 ((q p p ) + ) + x 4 ((q p 3 ) + ) 4 Page 8 of 93

131 CHAPTER 7. APPENDICES (x + x + x 3 + x 4 ) = 6 dx dx 3 dx 4 q m + + x (p q p )+ x 3 (p + p (q p + q p p p )) + x 4 (p 3 q p 3 ) 4 = 6 dx dx 3 dx 4 q m + + x p + x 3(p + p ) + x 4 p 3 (x p + x 3 (p + p ) + x 4 p 3 )q 4 = (7.79) 6 6 where we have dened dx dx 3 dx 4 l (x p + x 3 (p + p ) + x 4 p 3 ) + x 3 p p + dx dx 3 dx 4 (l ) 4 ; l q (x p + x 3 (p + p ) + x 4 p 3 ) ) 4 = l = q + (x p + x 3 (p + p ) + x 4 p 3 ) (x p + x 3 (p + p ) + x 4 p 3 )q (7.8) (x p + x 3 (p + p ) + x 4 p 3 ) x 3 p p : The momentum q can be wrtten to derent power as l = q (x p + x 3 (p + p ) + x 4 p 3 ) ) q = l + x p + x 3 (p + p ) + x 4 p 3 = l + ) q = l + + l ) (7.8) q 4 = l 4 + 4l 3 + 6l + 4l ; whch can be used to rewrte the numerator n the followng. The numerator can be wrtten as h Tr =q(=q p = )(=q p = p = )(=q p = ) = 4 3 (q ) q q p 3 + q q p 3 q p 3 q q p q+ q p q q q p + q p p 3 q p q p 3 + q p 3 q p q p q + q p q q q p + q p p 3 q p q p 3 + q p 3 q p q p q + q p q q p q + q p q p 3 q p p 3 + q p 3 p q + q p p q q p p q + q p q p p p 3 + q p p p 3 q p 3 p + q p p q q p p q + q p p q p p p 3 + q p p p 3 q p 3 p p = 4 h q 4 q p 3 q q q p + q p p 3 q p q + q p p 3 q p q + q p q p 3 q p p 3 + (7.8) q p p q p p p 3 + q p p p 3 q p 3 p p = 4 l 4 + l (6 p 3 p p + p p 3 + p p ) + l l ( p 3 4 p p + p p p 3 (p + p ) + p p 3 + p p 3 + p p p p p 3 p p p 3 p 3 p p = 4 l 4 + l A + l l B + C By nsertng the denomnator and numerator we get the ampltude M 4;top loop = t dx dx 3 dx 4 d d l () d l 4 + l A + l l B + C (l + ) 4 Page 9 of 93

132 CHAPTER 7. APPENDICES = t B g t dx dx 3 dx 4 3 d (4) + C # = 8 4 t " d(d + ) (4) d= 4 3 d + C dx dx 3 dx 4 (4) " (4) + nte! ; 4 d (4) d d (4) d 4 # d=4! log + log(4) + O() A d 3 d (4) A B g 3 d (7.83) where we have used the dmensonal regularzaton to regularze the ntegral and p s the total ncomng four momentum. We can determne the Hggs four-pont couplng renormalzaton constant Z from the counterterm condton as follows = (Z ) 3 = (M 4;scalar loop + M 4;top loop ) (4) + nte t (4) + nte 6 44 t (4) (4) + nte : = 9 ) Z = + (7.84) To calculatng L n Eq. (7.74), whch s needed to calculate the -functon, we need to calculate the wave functon renormalzaton constant Z n Eq. (7.7). The dagrams that contrbute to ths renormalzaton constant are the Hggs propagator correctons shown n Fgure 7.6. p p q q q q p q Fgure 7.6: Loop dagrams of the Hggs propagator where the rst dagram gves an one loop correctons to the couplng. Ampltude of the rst dagram wth a top-loop s M ;top loop = ( )3 t d p 4 p Tr[=p(=p = q)] d 4 p p(p () 4 (p + )((p q) + ) = q) 6 t () 4 (p + )((p q) + ) d 4 p p(p = 6 q) t dx () 4 dx (x (p + ) + x (p + q pq (x + )) + x ) d 4 p d 4 p = 6 t () 4 () 4 dx p(p q) (p + + x (q pq)) = 6 t dx p pq (l ) (7.85) Page 3 of 93

133 CHAPTER 7. APPENDICES = 6 t dx d d l () 4 l + (x x)q + (x x)q (4) d= where we have dened = 6 (l ) t ( d=) () d= ; dx d (4) d= d= ( d=) () l p x q ) l = p + x q pqx (x x)q We can expand the gamma functon (x) and the functon (= ) x around as follows ( d=) ' + and d= d= 4 = (4) (4) d= ' (4) h ( d=) log + log(4) ; (7.86) (7.87) whch gves us followng approxmaton d= ( d=) ' (4) d= (4) log + log(4) + : (7.88) By usng ths approxmaton we can wrte the ampltude of the top-loop dagram as M ;top loop ' 6 t dx = 3 6 t = 6 t (4) dx(x x)q (4) q + nte log + log(4) (4) + : : : = 8 t + (4) 6 log + log(4) (4) + : : : q (7.89) The ampltude of the other dagram wth a scalar loop can be wrtten as M ;scalar loop = ( 3 ) d 4 p () 4 p m H + = ( 3 ) d 4 l () 4 l = 3 d=! ( d=) ' 3 4 (4) d= 4 () (4) log + log(4) + = 3 4 (4) m H + nte (7.9) We can determne the wave functon renormalzaton constant from the counterterm condton as follows (q Z m ) = q (Z ) m hz m H = (M ;top loop + M ;scalar loop ) = 6 t (4) + nte ) Z = 6 t (4) + nte ; q 3 4 (4) m H + nte (7.9) where the dagram wth the scalar loop does not contrbute to the renormalzaton constant. Page 3 of 93

134 CHAPTER 7. APPENDICES Now, we can determne L by substtutng the renormalzaton constant Z and Z n Eq. (7.84) and Eq. (7.9) respectvely nto the rst equaton n Eq. (7.7), whch gves us that ln(z Z ) = 6 ln From ths we get + X n= L n ( t ; ) n (4) 44 t (4) = ln Z + ln Z = ln = 6 t (4) + 6 t (4) + 6 (4) 44 t (4) : (7.9) L = t 6 (4) + 44 t : (7.93) (4) (4) By substtutng L nto the rst equaton n Eq. (7.74), we get ( t ; ) t L (4) t (4) (4) 4 t + 4 (4) 4 t (7.94) = (4) t + t ; whch s the -functon of the Hggs four-pont couplng n the Standard Model to one loop order: ( t = (4) t + t : (7.95) Page 3 of 93

135 CHAPTER 7. APPENDICES Appendx F: The Scalar Sector and Vector Bosons n MWT In ths appendx, we derve the varous spnor blnears, the scalar M matrx and the vector A matrx n MWT theory n Chapter 3. Spnor Blnears Accordng to Ref. [] we have that 5 A ; C " " A and A ; (7.96) and the notaton for the spnors and ts adjont are U = (U L; ; U y _ R )T and U = (U R ; U y L; _ )T, respectvely. Therefore, we have that the spnor blnears are UU =U y R U L; + U y L; _ U _ R DD =D y R D L; + D y L; _ D _ R UD =U y R D L; + U y L; _ D _ R DU =D y R U L; + D y L; _ U _ R U 5 U = U y R ; U L; _ = U y R U L + U y L _ U _ R D 5 D = D y R D L + D y L _ D _ R D 5 U = D y R U L + D y L _ U _ R U 5 D = U y R D L + U y L _ D _ R A U T CU = U L; ; U R " " = U L; U L U _ RU R; _ D T CD = D L; D L D _ RD R; _ U T CD = U L; D L U _ RD R; U L U _ R U L U _ R A = U y R ; U y L U L U _ R A = UL; ; U _ U L A U R; _ A D T CU = D L; U L D _ RU R; _ U T C 5 U = U L; ; U R " " U L; U _ R A = UL; ; U _ U L U R; _ A =U L; U L U _ RU R; _ D T C 5 D =D L; D L D _ RD R; _ U T C 5 D =U L; D L U _ RD R; _ Page 33 of 93

136 CHAPTER 7. APPENDICES D T C 5 U =D L; U L D _ RU R; _ U U = U y R ; U _ L; U L U _ R =U y R U R + U y _ L U _ L = U y R U R UL U y L D D =D y R D R DL _ Dy _ L U D =U y R D R DL U y L D U =D y R U R UL _ Dy _ L U 5 U = U y R ; U _ L; =U y R _ U _ R + U L _ U y _ L D 5 D =D y R D R + D L _ Dy _ L U 5 D =U y R D R + D L U y L D 5 U =D y R U R + U L _ Dy _ L U T C U = U L; ; U " R " A A = U y R _ U _ R + U y L; U U L; U _ R U L; U _ R =U L _ U _ R + U R; U L; = U L _ U _ R D L _ U _ R D T C D =D L _ D _ R D L _ D _ R U T C D =U L _ D _ R D L _ U _ R D T C U =D L _ U _ R U L _ D _ R D T C 5 D =D L _ D _ R + D L _ D _ R U T C 5 D =U L _ D _ R + D L _ U _ R D T C 5 U =D L _ U _ R + U L _ D _ R U T C 5 U = U L; ; U _ " " A A = U y R _ U _ R U y L; U L; U L; U _ R =U L _ U _ R U R; U L; = U L _ U _ R + U L _ U _ R = U L _ U _ R D T C 5 D =D L _ D _ R + D L _ D _ R U T C 5 D =U L _ D _ R + D L _ U _ R D T C 5 U =D L _ U _ R + U L _ D _ R A Page 34 of 93

137 CHAPTER 7. APPENDICES The Scalar Sector: Now, we wll derve the scalar M matrx. The charge egenstates are v + H UU + DD; ( U 5 U + D 5 D) A ~ 3 UU DD; 3 ( U 5 U D 5 D) A + ~ p ~ DU; + p D 5 U (7.97) A ~ + p ~ UD; + p U 5 D for the technmesons, and UU U T CU DD D T CD UD 8 + p 9 U T CD ~ UU ~ 4 + ~ 5 + ~ 6 + ~ 7 U T C 5 U ~ DD ~ 4 + ~ 5 ~ 6 ~ 7 D T C 5 D ~ UD ~ 8 + ~ 9 p U T C 5 D (7.98) for the technbaryons. The varous elements of the M matrx n terms of the blnears n Appendx F are h M =UL U L; = 4 + ~ 4 + ( 5 + ~ 5 ) ~ 6 + ( 7 + ~ 7 ) h M =M = UL D L; = DLU L; = 8 + ~ 8 + ( 9 + ~ 9 ) h ~ 3 M 3 =M 3 = (U y R ) U L; = U L (U y R ) = M 4 =M 4 = (D y R ) U L; = U L (Dy R ) = M =D LD L; = h + ~ ( + ~ ) h 4 + ~ 4 + ( 5 + ~ 5 ) 6 ~ 6 ( 7 + ~ 7 ) h M 3 =M 3 = (U y R ) D L; = DL (U y R ) = + ~ + ( + ~ ) M 4 =M 4 = (D y R ) D L; = DL (Dy R ) = h + 3 ~ 3 M 33 =(U y R ) (U y R ) = M 43 =M 34 = (D y R ) (U y R ) = (U y R ) (D y R ) = M 44 =(D y R ) (D y R ) = h 4 + ~ 4 ( 5 + ~ 5 ) ~ 6 ( 7 + ~ 7 ) h 8 + ~ 8 ( 9 + ~ 9 ) h 4 + ~ 4 ( 5 + ~ 5 ) 6 ~ 6 + ( 7 + ~ 7 ) ; (7.99) Page 35 of 93

138 CHAPTER 7. APPENDICES where we have used that UU + DD =U y R U L; + U y L; _ U _ R + Dy R D L; + D y L; _ D _ R = M 3 + M y 3 + M 4 + M y 4 = UU DD =U y R U L; + U y L; _ U _ R D y R D L; D y L; _ D _ R = M 3 + M y 3 M 4 M y 4 = ~ 3 ~ 3 A DU =D y R U L; + D y L; _ U _ R = M 4 + M y 3 = ~ ~ ~ ~ p A + UD =U y R D L; + U y L; _ D R _ = M 3 + M y 4 = ~ + ~ ~ + p ~ A ( U 5 U + D 5 D) = U y R U L; + U y L; _ D y R D L; + D y L; _ D _ R = M 3 + M y 3 M 4 + M y 4 = ( U 5 U D 5 D) =( U y R U L; + U y L; _ U _ R + Dy R D L; D y L; _ D _ R ) = ( M 3 + M y 3 + M 4 M y 4 ) = 3 3 D 5 U =( D y R U L; + D y L; _ U _ R ) = ( M 4 + M y 3 ) = p + U 5 D =( U y R D L; + U y L; _ D _ R ) = ( M 3 + M y 4 ) = + + p UU D T CD = D L; DL D RD _ R; _ = M M y 44 = DD U T CU = U L; U L U _ RU R; _ = M M y 33 = U T CD = U L; D L U _ RD R; _ = M M y 43 = p UD U T C 5 U =[U L; U L U _ RU R; _ ] = [M + M y 33 ] = ( ~ 4 + ~ 5 + ~ 6 + ~ 7 ) ~ 4 + ~ 5 + ~ 6 + ~ 7 D T C 5 D =[D L; D L D _ RD R; _ ] = [M + M y 44 ] = ( ~ 4 + ~ 5 ~ 6 ~ 7 ) ~ 4 + ~ 5 ~ 6 ~ 7 U T C 5 D =[U L; D L U _ RD R; _ ] = (M + M y 34 ) = ( ~ 8 + ~ 9 ) ~ 8 + ~ 9 p ~ UD : Therefore, we can wrte the scalar charge egenstates as follows ~ UU ~ DD = ( UU + DD); A = ( UU DD); = ( U 5 U + D 5 D); = ( U 5 U D 5 D); A + = p DU; + = p D 5 U; A = p UD; = p U 5 D (7.) The M matrx can be wrtten n the form M = Q Q " = UU = U T CU; ~ UU = U T C 5 U; DD = DT CD; ~ DD = DT C 5 D; UD = p U T CD; ~ UD = p U T C 5 D U L U L; U L D L; U L U R; U L D R; D L U L; D L D L; D L U R; D L D R; U R U L; U R D L; U R U R; U R D R; D R U L; D R D L; D R U R; D R D R; C A (7.) : (7.) Page 36 of 93

139 CHAPTER 7. APPENDICES There, we have that M = because UU + ~ UD+ UD ~ UU p + + +A + +A + p UD+ p UD ~ DD + ~ +A DD p + A + + +A + +A + p + A p+a UU + ~ UD + ~ p UD UU UD + ~ UD p DD + ~ DD C A ; (7.3) UU + ~ UU = UD + ~ UD p = U T CU U T C 5 U = U L; U L = U L U L; = M U T CD U T C 5 D = U L; DL = U L D L; = DLU L; = M = M A = ( UU + DD U 5 U D 5 D) ( U 5 U D 5 D UU + DD) + + A + p = =U L U y R; = U y R U L; = M 3 = M 3 D 5 U + DU = U L D y R; = Dy R U L; = M 4 = M 4 DD + ~ DD = + A p = D T CD D T C 5 D = D LD L; = M U 5 D + UD = D LU y R; = U y R D L; = M 3 = M 3 + A = ( UU + DD U 5 U D 5 D) + ( U 5 U D 5 D UU + DD) (7.4) =D LD y R; = Dy R D L; = M 4 = M 4 UU + ~ UU =U y R U y R; = M 33 UD + ~ UD p =U y R Dy R; = Dy R U y L; = M 34 = M 43 DD + ~ DD =D y R Dy L; = M 44: Accordng to Ref. [] we have when we take the charge conjugated of a spnor, then we make the U L; The Vector Bosons: U _ R U y R; U _ L A : (7.5) We wll now derve the vector A matrx. The relatons between the charge egenstates and the wavefunctons of the composte objects are v A 3 U U D D; a A 9 U 5 U D 5 D; v + A A p D U; a + A7 A 8 p D 5 U; v A + A p U D; a A7 + A 8 p U 5 D; (7.6) v 4 A 4 U U + D D; Page 37 of 93

140 CHAPTER 7. APPENDICES for the vector mesons, and x UU A + A + A + A 3 x DD A + A A A 3 x UD A4 + A 5 p D T C 5 U U T C 5 U D T C 5 D (7.7) s UD A6 A 5 p U T C D for the vector baryons. We can wrte the varous elements of the A wth the blnears n Appendx F A =U L _ U _ L 4 Q k _ Q _ ;k = p (A3 + A 4 + A 9 ) A =D L _ U _ L = p (A A + A 7 A 8 ) A 3 =U R _ U _ L = p (A + A + A + A 3 ) A 4 =D R _ U _ L = p ( A5 + A 6 + A 4 + A 5 ) A =U L _ D _ L = p (A + A + A 7 + A 8 ) A =D L _ D _ L 4 Q k _ Q _ ;k = p ( A3 + A 4 A 9 ) A 3 =U R _ D _ L = p (A5 A 6 + A 4 + A 5 ) A 4 =D R _ D _ L = p ( A5 + A 6 + A 4 + A 5 ) A 3 =U L _ U _ R = p (A A + A A 3 ) A 3 =D L _ U _ R = p ( A5 A 6 + A 4 A 5 ) A 3 3 =U R _ U _ R 4 Q k _ Q _ ;k = p ( A3 A 4 + A 9 ) A 4 3 =D R _ U _ R = p ( A A + A 7 + A 8 ) A 4 =U L _ D _ R = p (A5 + A 6 + A 4 A 5 ) A 4 =D L _ D _ R = p (A A A + A 3 ) A 3 4 =U R _ D _ R = p ( A + A + A 7 A 8 ) A 4 4 =U R _ D _ R 4 Q k _ Q _ ;k = p (A3 A 4 A 9 ); (7.8) whch gves that the blnears are U U D D =U R U R UL U L D R D R + D L _ D _ L =A 3 3 A A A = p A 3 A 3 v D U =D R U R UL _ D _ L = A3 4 A = p (A A ) p (A A ) v + Page 38 of 93

141 CHAPTER 7. APPENDICES U D =U R D R DL U L = A4 3 A = p (A + A ) p (A + A ) v U U + D D =U R U R UL U L + D R D R DL _ D _ L p =A 3 3 A + A4 4 A = A 4 A 4 v 4 U 5 U D 5 D =U R U R + U L U L D R D R DL _ D _ L =A A A 4 4 A = pa 9 A 9 a D 5 U =D R U R + U L _ D _ L = A3 4 + A = p (A 7 A 8 ) p (A 7 A 8 ) a + U 5 D =U R D R + D L U L = A4 3 + A = p (A 7 + A 8 ) p (A 7 + A 8 ) a U T C 5 U =UL U R + U L U R = A3 = p (A + A + A + A 3 ) (7.9) (A + A + A + A 3 ) x UU D T C 5 D =DL D R = A4 = p (A + A A A 3 ) (A + A A A 3 ) x DD D T C 5 U =DL U R + U L D R = A3 + A4 = p (A 4 + A 5 ) x UD U T C D =UL D R DL U R = A4 A 3 = p (A 6 A 5 ) s UD : Therefore, we obtan that the charge egenstates are v = p ( U U D D); a = p ( U 5 U D 5 D); v + = D U; a + = D 5 U; v = U D; a = U 5 D; v 4 = p ( U U + D D) (7.) The A matrx can be wrtten n the form: x UU = p U T C 5 U; x DD = p D T C 5 D; x UD = U T C 5 D; s UD = U T C 5 D: (7.) A ;j =Q _ Q ;j _ 4 j Q k _ Q ;k _ = UL U L DL U L UL _ D _ L DL _ D _ L UL U R DL U R UL D R DL D R U R _ U _ L U R _ D _ L U R _ U _ R U R _ D _ R D R _ U _ L U R _ D _ L D R _ U _ R D R _ D _ R C A 4 j Q k _ Q ;k _ : (7.) Page 39 of 93

142 CHAPTER 7. APPENDICES Fnally, we can wrte the A matrx as A = a +v +v 4 a +v a + +v + a p v +v 4 x x puu UD s UD x UD +s UD x p UU x UD s UD a v v 4 p x pdd a + v + x UD +s UD x p DD a v a +v p v 4 C A ; (7.3) where a + v p + v 4 a + + v + = 4 U 5 U D 5 D ( U U D D) ( U U + D D) = 4 U R _ U _ R + U L _ U L (D R _ D _ U L _ U L D R _ D _ R = U L _ U L 4 U L _ U L = U L _ U _ L 4 Q k _ Q _ ;k ; R + D L _ D L ) U R _ U _ R + + D L _ D L + U R U R + = D 5 U D U = D R _ U _ R + U L _ D _ L D R _ U _ R + U L _ D _ L = U L _ D _ L ; (7.4) x UU p = U T C 5 U = U L _ U _ R etc. Page 4 of 93

143 CHAPTER 7. APPENDICES Appendx G: Dscrete transformatons of spnors, Q vector and M matrx In ths appendx we wll derve the dscrete transformatons (party, charge conjugaton and CP transformatons) of spnors, the Q vector n eg. (3.7) and the M matrx n Eq. (7.). We have that a Drac spnor transforms under party as follows (acc. Eq. (4.5) n Ref. []) P U (x)p U L;(x) = U _ R U _ R (Px) U L; (Px) A P = U (Px) U L;(Px) U _ R (Px) A : (7.5) Therefore we have that the left- and rght-handed Weyl spnors transform under party as follows A P U L; (x)p =U R _ (Px) P U R _ (x)p =U L;(Px); (7.6) and f we take the complex conjugated of them then we get P P U y _ L (x)p = U y R; (Px) (7.7) U y y _ R; (x)p =UL (Px): We can use these transformatons to wrte a party transformaton expresson of the Q vector. The Q vector party transforms as follows P Q A P = P U L; (x)p P D L; (x)p P U R; (x)p P D R; (x)p C A = U _ R (Px) D _ R (Px) U _ L (Px) D _ L (Px) C A = Q B; _ E + AB : (7.8) The charge conjugaton transformaton of a Drac spnor (acc. Eq. (4.4) n Ref. []) can be wrtten as C U (x)c U L;(Px) U _ R (Px) " " " " A A A C = C U T = C(U y ) T 4 U y L; _ U y U y R U y L; A U y R; U y _ L Therefore we can wrte the C transformatons of the left- and rght-handed Weyl spnors as follows A : 3 A5 T (7.9) C U L; C =U y R; C U _ RC =U y _ L ; (7.) Page 4 of 93

144 CHAPTER 7. APPENDICES and by takng the complex conjugaton of them then we get C U y L; _ C =U R; _ : (7.) C U y R C =U L We can use these C transformatons to wrte the C transformaton expresson of the Q vector whch s C Q A C = C U L; C C D L; C C U R; C C D R; C C A = U R; D R; U L; D L; C A = Q B E + AB : (7.) By usng the P and C transformatons expressons we can derve an expresson for how the M matrx n Eq. (7.) transforms under P and C transformatons. The P transformaton of M s P M AB (x)p =P Q A (x)q B;(x)P = P Q A (x)p P Q B; (x)p = Q C; _ (Px)E+ _ ACQ D (Px)E+ BD = Q _ C; _ (Px)Q D (Px)E+ AC E+ BD =Q _ C (Px)Q D; _ (Px)E+ AC E+ BD = (EM E) AB ; (7.3) and the C transformaton of M s C M AB C =C Q AQ B; C = C Q ACC Q B; C = Q CQ D; E AC + E+ BD = (EME) AB: (7.4) We can also make a CP transformaton of the M matrx by combnng the P and C transformatons of M from the two prevous equatons. Thus, the CP transformaton of M s (CP ) M AB CP =P (EME) AB P = (EEM EE) AB = M AB (7.5) Page 4 of 93

145 CHAPTER 7. APPENDICES Appendx H: Wtten Anomaly We have that an SU() gauge theory s mathematcally nconsstent f there are an odd number of lefthanded fermon doublets and no other representatons n ths theory. The begnnng pont s that the fourth homotopy group of SU() s nontrval because 4 (SU()) = Z. Ths means that there s a gauge transformaton U (x) n four-dmensonal eucldean space, whch "wraps" around the gauge group such that t can not be contnuously deformed to the dentty. The meanng of that the homotopy group s equal Z s that a gauge transformaton that wraps twce around the SU() group can be deformed to the dentty. To begn wth we can wrte the eucldean path ntegral for the free gauge eld A wthout fermons DA exp g d 4 x Tr(F F ) : (7.6) In ths path ntegral we are double countng because for every gauge eld A, there s a gauge transformed gauge eld A U = U A U U: (7.7) Wthout fermons n the theory, the double countng cancels out when one calculates vacuum expectaton values. By ncludng a sngle doublet of left-handed fermons, we now have the path ntegral Z = DA D D exp We would lke to ntegrate out the Drac fermons whch gves D D exp d 4 x g Tr(F F ) + D : (7.8) d 4 x D = Det( D ): (7.9) Here the rght-hand sde s the nnte product of all egenvalues of the operator D. Now, wth the gauge group SU(), a doublet of Drac fermons s exactly the same as two left-handed or Weyl doublets. Therefore the ntegraton of the fermon n Eq. (7.9) wth a sngle Weyl doublet would gve the square root of Det( D ). Thus for the sngle Weyl doublet, the partton functon s Z = DA D D Det( D ) = exp An ambguty arses here, because the square root has two sgns. d 4 x g Tr(F F ) : (7.3) There s nothng that guarantees that Det( D ) = s nvarant under the topologcally non-trval gauge transformaton U. Actually, Det( D ) = s odd under U. We can show that for any gauge eld A, that Det[ D (A )] = = Det[ D (A U )]= : (7.3) I.e. f we vary the gauge eld A to A U contnuously, then we can end up wth the opposte sgn of the square root. It s elaborated n more detal n Wtten's own artcle about ths SU() anomaly Ref. [9]. The consequence of ths s that the partton functon n Eq. (7.3) vanshes dentcally, because the contrbuton of any gauge eld A s exactly cancelled by transformed gauge eld A U wth opposte sgn. Page 43 of 93

146 CHAPTER 7. APPENDICES Ths gves problems when we calculate the path ntegral Z X wth nserton of any gauge nvarant operator X whch s dentcally zero. The expectaton values are ndetermnate, because hx = Z X =Z = =. Therefore, the theory s ll-dened. Now let us consder some generalzatons. If we have n left-handed fermon doublets, then the ntegraton would gve [Det( D )] n=. If n s even, then the sgn of the square root does nothng, but f n s odd, then the theory s nconsstent as before. Ths perssts even f we add addtonal gauge or Yukawa couplngs to the SU() gauge theory. Example the Standard Model of strong, weak and electromagnetc nteracton wth the gauge group SU(3)C SU()W U()Y would be nconsstent f the number of lefthanded fermon doublets were odd. Ths s not the case n the SM, because there s a lepton left-handed doublet for each quark doublet, and therefore the number of left-handed doublets s even. Fnally, we can consder other gauge groups than SU(), we have 4 (SU(N )) = ; N > ; 4 (O(N)) = ; N > 5; (7.3) 4 (Sp(N )) = Z ; any N: Thus the non-trval condtons arse only for Sp(N) gauge groups. In concluson, the Wtten anomaly arses when the number of left-handed doublets s odd, and t s a problem exclusvely applyng to Sp (N ) gauge groups, SU() group because SU() Sp(), and O(N < 6), except for O(). Page 44 of 93

147 CHAPTER 7. APPENDICES Appendx I: The Electroweak Precson Parameters The EW precson parameters called S and T descrbe how much the EW symmetry and custodal symmetry are broken, respectvely. We have the followng dentons (Ref. [43]) ^S W 3 B (); ^T g m [ W 3 W 3() W + W ()]; W W g m W Y g m W W 3 W 3(); BB (); (7.33) where VV (q ) wth V V = fw 3 B; W 3 W 3 ; W + W ; BBg are the self-energy of the vector bosons shown n Fgure 7.7. The partcles and are the partcles n the SM and beyond the SM, respectvely, that run n the loop and couple to the EW gauge bosons. The dervatve wth respect to q of the self-energy s denoted wth a prme. The Peskn-Takeuch parameters S and T are related to the new ones above va (Ref. [43]) S 4s = ^S Y W; T = ^T s W Y; (7.34) W s W where s the electromagnetc structure constant and s W s the weak mxng angle. Data from the LEP experments set the EW parameters to be (Eq. (.7) n Ref. [73]) S = :5 :; T = :8 :; (7.35) U = : : where the uncertantes are from the nputs. These parameters are n excellent agreement wth the SM values of zero. Values of these parameters derent from zero are due to new physcs. The T parameter s related to the parameter, whch s a measure for how much the custodal symmetry s broken. The relaton between these two parameters (cf. Eq. (.68) n Ref. [73]) s = (m Z )T ' + (m Z)T: (7.36) The value of the parameter accordng to (m Z ) = 7:95 :7 (cf. Eq. (64) n Ref. [44]) and the value of the T parameter n Eq. (7.37) s = :6 :9. V V V V Fgure 7.7: The loop dagrams whch gve rse to the self-energy for the vector bosons V and V. By Fxng U = (as s also done n Fgure 7.8) moves S and T slghtly upwards (cf. Eq. (.73) n Page 45 of 93

148 CHAPTER 7. APPENDICES Ref. [73]) S = :7 :8; T = : :7: (7.37) Fgure. s Fgure.6 n Ref. [73]. We have that the black dot ndcates the SM values S = T =, whle the red ellpse llustrates constrants on S and T (for U = ). The black dot that ndcates the SM values s nsde constrants, and therefore the SM precson s good. If one of the EW parameters ncreases/decreases, t s needed to ncrease/decrease the other parameter. Fgure 7.8: constrants on S and T (for U = ) from varous nputs combned wth m Z. The black dot ndcates the SM values S = T =. The gure s Fgure.6 n Ref. [73]. Page 46 of 93