investigating a possible dynamical origin of the...

Investigating a Possible Dynamical Origin of

the Electroweak Scale

University of Southern Denmark

Master's Thesis in Particle Physics

Martin Rosenlyst Jørgensen

CP3-Origins

Supervised by

Ass. Prof. Mads Toudal Frandsen, CP3-Origins

Postdoc Tommi Alanne, CP3-Origins

June 1, 2017

Abstract

One era came to an end in July 2012 when two experiments, CMS and ATLAS, at the Large Hadron

Collider at CERN announced the discovery of a new resonance consistent with the Standard Model Higgs

boson. The Higgs boson was the last missing piece of the Standard Model of elementary particle physics,

our most fundamental description of the elementary particles and their interactions via three of the four

forces, the exception being gravity. Although the Standard Model is in agreement with a great number

of experimental measurements, it cannot explain all the observations.

The thesis begins with an introduction to the Standard Model, the reasons why there must be some

more fundamental theory beyond the Standard Model. This thesis will elucidate extensions of the Stan-

dard Model, where the Higgs sector is replaced by a strongly interacting sector. We will focus mostly on

the naturalness i.e. the problem that the mass of the Higgs boson is very �ne-tuned.

Therefore, in this thesis we will investigate extensions of the Standard Model, where the standard

Higgs sector is replaced by a strongly interacting sector. After we have developed the tools to study

strongly interacting theories, we will discuss and develop three concrete examples: the Minimal Walking

Technicolor (MWT) model, a Composite Higgs (CH) model and a Partially Composite Higgs (PCH)

model. We will investigate the vacuum stability of the PCH model by calculating the running of a new

fundamental scalar self-coupling, and we discover that this kind of models are �ne-tuned and the vacuum

is unstable for a large part of the parameter space. This part of the thesis is novel research.

Acknowledgements

This master's thesis is done at the Centre for Cosmology and Particle Physics Phenomenology (CP3-

Origins), University of Southern Denmark. I would like to thank my supervisors Mads Toudal Frandsen

and Tommi Alanne for all the guidance. I would also like to thank my fellow student Mette L. A.

Kristensen who took time to discuss the thesis with me. Finally, I would like express my deepest gratitude

to Sophie and my family for their support, patience and love.

Contents

1 Introduction 4

2 Introduction to Elementary Particle Physics 6

2.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Realization of symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Unitarity of WLWL Scattering Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Custodial Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.1 Custodial Symmetry at Tree Level . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.2 Custodial Symmetry at Loop Level . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Triviality and Vacuum Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5.1 Triviality of QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5.2 Triviality of Higgs Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.3 Vacuum Stability in the SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.6 Higgs Mass Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.7 The EW Hierarchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.7.1 Fine-Tuning of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.7.2 Fine-Tuning of the Higgs Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.8 Chiral Symmetry Breaking in QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.8.1 Quantum Chromodynamics (QCD) . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.8.2 Construction of an E�ective Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . 39

2.8.3 Chiral Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.9 Technicolor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.9.1 Simple Technicolor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.10 Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3 Minimal Walking Technicolor 49

3.1 The Underlying Lagrangian for Minimal Walking Technicolor . . . . . . . . . . . . . . . . 50

3.2 Low Energy Theory for MWT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2

CONTENTS

3.2.1 Composite Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.2 Composite Vector Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2.3 Fermions in the E�ective Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2.4 Yukawa Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3 Extended Technicolor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.4 Walking Technicolor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.5 Weinberg Sum Rules and the S Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 77


4 Composite Higgs Dynamics 81

4.1 The Fundamental Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.2 Electroweak Vacuum Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2.1 The �B Vacuum: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2.2 The �H Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2.3 A Superposition of the two Vacua: . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3 Loop Induced Higgs Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3.1 Gauge Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3.2 Top Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.3.3 Explicit Breaking of SU(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.4 Fine-Tuning of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94


5 Partially Composite Higgs Dynamics 96

5.1 The Fundamental Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2 Construction of the E�ective Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.3 The Vacuum Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.4 Scalar Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.5 The Normalization Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.6 The Angles in the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.7 The Parameter Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.8 The Vacuum Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107


6 Conclusions 112

7 Appendices 114

Page 3 of 193

Chapter 1

Introduction

The Large Hadron Collider (LHC) is the biggest scienti�c instrument ever created. It accelerates and

collides protons along the 27 kilometers long tunnel excavated beneath the French-Swiss border. The main

physics goal of the LHC is to determine the origin of electroweak symmetry breaking, i.e. the mehanism

providing the mass for the elementary particles. The �rst step towards this goal was taken when the

CMS and ATLAS collaborations (Refs. [6,7]) announced that they had discovered a new resonance with

properties consistent with those of the Standard Model Higgs, within the measurement uncertainties.

The Higgs boson was up until 2012 the missing piece of the Standard Model (SM)1 is responsible for the

origin of mass of the elementary particles in that model, for curing the would-be violation of unitarity

in the weak sector and to bring agreement between the predicted electroweak precision observables and

the measured at Large Electron�Positron Collider (LEP) experiments.2 The next goal is to measure and

investigate what lies beyond the SM. Despite of all successes of the SM it cannot explain all current

observations including neutrino masses, baryogenesis and dark matter, and there are a various reasons

that it is not the most fundamental theory of Nature.

One important reason that the Standard Model may not be a complete theory of electroweak (EW)

symmetry breaking is that the mass of the Higgs boson is very �ne-tuned. The electroweak energy scale

is namely 17 orders of magnitude smaller than the Planck energy scale that characterizes gravity. It

results in a naturalness problem of the electroweak scale which is known as the electroweak hierarchy

problem, because it does not seem natural that the mass is extreme �ne-tuned. In this thesis, our main

motivation is to search after a possible dynamical origin of the electroweak scale which would be natural.

Other issues with the SM Higgs sector are the triviality problem and problems in �avor physics. These

further motivate the quest for a theory of EW symmetry breaking beyond the SM Higgs model.

The Naturalness paradigm will be adressed in composite formulations of the Higgs mechanism, includ-

ing so-called technicolor (TC) models, composite Higgs (CH) models, bosonic technicolor (BTC) models

1The �rst step towards the Standard Model was the discovery in 1961 of a way to combine the electromagnetic andweak interactions discovered by Sheldon Glashow (Ref. [62]). In 1967 Steven Weinberg and Abdus Salam incorporated theHiggs mechanism (Refs. [59�61]) into Glashow's electroweak interaction giving it its modern form (Refs. [63, 64]).

2LEP collided electrons with positrons at energies that reached 209 GeV (cf. Ref. [8]). In 2001 it was shut down tomake way for the LHC, which reused the LEP tunnel.

4

CHAPTER 1. INTRODUCTION

and partially composite Higgs models (PCH). The main idea is to have techniquarks and technigluons

analogous to the quarks and gluons as in quantum chromodynamics (QCD), that con�ne in technihadrons

(technimeson and technibaryons) after chiral symmetry breaking. This con�nement and chiral symmetry

breaking provides a natural dynamical origin of the electroweak scale. By introducing technicolor the

Higgs mechanism has a natural scale and is non-trivial, but it still does not explain the �avor physics.

The TC models itself has no mechanism that explains the origin of SM fermion masses. For that we

would introduce extended technicolor (ETC). Such ETC models cause their own set of problems. It is

challenging to generate enough mass to the heaviest fermions in some realizations it is already problematic

to produce the mass of the charm quark. Simultaneously, ETC contributes to the �avor changing neutral

currents (FCNC) and contributes to discrepancies with precision electroweak measurements. The primary

solution to these potential problems is to assume that the TC dynamics is distinctly unlike QCD. This

scenario is referred to as walking technicolor (walking TC), where the coupling constant of TC evolves

slowly across a large energy scale as opposed to the 'running' coupling constant in QCD.

Another issue in these TC models is that it is heard to explain the mass of the observed 125 GeV boson

at LHC. In TC the Higgs boson is identi�ed with the lightest scalar resonance, the techni-� (similar to the

� resonance in QCD). By rescaling this resonance in QCD to technicolor, it is too heavy to be the observed,

unless the number of technicolors is very high (cf. Ref. [11]). This in turn is constrained by electroweak

precision measurements. This issue is alleviated by CH and PCH (BTC in the TC limit) models, where

the Higgs boson is identi�ed with a composite Goldstone boson and a mixture of a composite Goldstone

boson (scalar excitation in the TC limit) and a fundamental scalar, respectively. Unfortunately, the

parameters in both kind of models end up �ne-tuned. By performing a novel computation of the vacuum

stability in the PCH model in Ref. [3], we demonstrate that its vacuum alignment angle seems �ne-tuned.

This thesis consists of a chapter that gives an introduction to the SM of elementary particle physics

and its problems, and three chapters that discuss explicit extensions of the SM Higgs sector �rst presented

in the three research papers in Refs. [1�3], respectively. This thesis is organized as follows: In Chapter

2, the Standard Model, its vacuum stability, and its problems mentioned above are discussed, and how

these issues are addressed by a simple TC model which is a rescaled QCD model. The Chapter begins

with a discussion about the symmetries and why the symmetries along with renormalizability are the

primary reason for the predictive power of the Standard Model followed by a schematic review of the

Standard Model. The chapter ends with a discussion how the problems of EW symmetry breaking in the

Standard model are addressed in a simple TC model which is a scaled up version of QCD. In Chapter

3, the minimal walking technicolor (MWT) model in Ref. [1] is constructed, and we review ETC and

walking TC. Chapter 4 introduces CH models (following mostly Ref. [2]) by aligning the vacuum in

another direction away from the TC vacuum with the motivation to achieve a light Higgs boson from a

Goldstone Boson of the strong dynamics. In Chapter 5, the potential �ne-tuning problems in the CH

models are addressed by introducing a PCH model as in Ref. [3], where the Higgs boson is partially

composite and fundamental. We �nally present a novel analysis of the vacuum stability in this model

and the consequences for the viable parameter space of the model.

Page 5 of 193

Chapter 2

Introduction to Elementary Particle

Physics

In this chapter, the Standard Model of particle physics and its problems are discussed. The chapter

begins with a discussion of symmetries along with renormalizability and why the symmetries are the

primary reason for the predictive power of the Standard Model. Following by a schematic review of the

Standard Model and a discussion of each term in its total Lagrangian. We initiate a discussion of the

possible problems of the Standard Model including the unitarity ofWLWL scattering, custodial symmetry,

triviality and vacuum stability. By calculating the mass corrections to the mass of the Higgs boson, we

will �nd out that the Higgs mass is very �ne-tuned, which gives us a naturalness problem as de�ned and

quanti�ed by 't Hooft in Ref. [11]. The naturalness problem of the Higgs mass is called the electroweak

(EW) hierarchy problem, because the mass is 17 orders of magnitude smaller than the Planck mass that

characterizes gravity.

At the end of the chapter, we introduce chiral symmetry breaking in quantum chromodynamics

(QCD) and gives an introduction to a simple Technicolor model which is motivated by trying to address

the naturalness problem of the Higgs mass.

2.1 Symmetries

Let us start to discuss the questions, what is a symmetry of a particle physics model, and what is the

importance of these symmetries in particle physics?

A Symmetry means an invariance under a set of transformations. An popular example is a symmetric

geometric object which looks the same, if the object is rotated by an angle. The set of all symmetry

transformations form a symmetry group of the object. A rotation is called a continuous transformation,

while for example a re�ection transformation of the object that keeps the object invariant is called a

discrete transformation.

6

CHAPTER 2. INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS

The laws of Nature can be symmetric. This means that the form of the equation describing the law is

maintained under a change of space-time coordinates and/or variables. We can categorize the symmetries

such that geometric symmetries act on space-time coordinates and internal symmetries do not.

The continuous symmetries of the equation of motions can be related to conserved quantities. This is

quanti�ed by the Noether's theorem (the precise mathematical formulation is given in Ref. [20]), which

applies to both geometric and internal symmetries.

Noether's Theorem: If the equations of motion are invariant under a continuous transformation

with n parameters, there exist n conserved quantities.

If the equations of motion are invariant under translation in time, translation in space and rotations

in space, the corresponding conserved quantities are shown in the three �rst rows in Table 2.1. In the case

where we have relativistic particles, it is convenient to introduce Minkowski spaceM which is a real four-

dimensional vector space with the vectors x� = (ct; ~x) and with the metric (ds)2 = dx�dx� = (dt)2�(d~x)2.A semi-direct product of the Lorentz transformations x ! x0 = �x and the translations in space-time

x ! x0 = x + a (with a 2 M) form the Poincaré group, which leave the Minkowski metric invariant.

An elementary particle should not depend on its position in space-time or if the observer is in uniform

motion relative to it. Therefore, the Lagrangian describing the particle and its interactions should be

invariant under the Poincaré group.

Continuous Invariance Conserved Quantity

Time invariance Energy Conservation

Translation invariance Momentum conservation

Rotational invariance Angular momentum conservation

Gauge invariance Charge conservation

Table 2.1: Some symmetries and the associated conservation laws.

We can write an invariant Lagrangian under Poincaré transformations

LK = i �@� ; (2.1)

which is the kinetic term of a Dirac fermion (x). This term is invariant under a global U(1) phase

transformation

(x)! exp(ie�) (x); (2.2)

where e and � are space-time independent constants. If � is space-time dependent, Eq. (2.1) is no longer

invariant under the U(1) transformation. The term can be invariant by replacing the partial derivative

with the covariant derivative

@� ! D� = @� + ieA�; (2.3)

Page 7 of 193


where the gauge �eld A� transforms as

A� ! A� � @��(x): (2.4)

This procedure is called gauging and it �xes the form of interactions between the Dirac �eld and the

gauge �eld A�. The local phase transformations are called gauge transformations, and these kind of

theories are called Abelian gauge theories. The local and also the global U(1) symmetry are continuous

symmetries and the corresponding conserved quantities are the electric charge of the Dirac spinor (x)

and the number of particles, respectively.

This can be generalized to non-Abelian compact Lie gauge groups. In this case, all �elds carry an

additional index, i or a, which indicates the charge with respect to a gauge group for the fundamental and

the adjoint representation, respectively (Group representations are discussed in Appendix C). In these

theories, a gauge �eld can now be written as A� = Aa�TaA with the non-commuting T aA generators in the

adjoint representation of the gauge group, and Aa� are the component �elds of the gauge �eld for each

charge. The gauge transformation of a fermion �eld is thus

(x)! exp(i�aTaA) (x) � g (x); (2.5)

where �a are arbitrary functions, and a takes the same values as for the gauge �elds. The corresponding

covariant derivative is

D� = @� + ieAiTi (2.6)

with the Ti generators in the fundamental representation of the gauge group (in principle they can also

be in the adjoint representation instead). The corresponding gauge transformation for the gauge �elds

has then to take the inhomogeneous form in contrast to the Abelian theory in Eq. (2.4)

A� ! gA�g�1 + g@�g

�1: (2.7)

The di�erence between an Abelian gauge and an non-Abelian theories is that the generators of the gauge

group are commuting and not commuting, respectively. The gauge symmetry is an internal symmetry,

and the space-time and the internal symmetries are described in terms of Lie groups.

Another type of symmetries than the Lorentz and the internal symmetries are the discrete symmetries.

A discrete symmetry is a symmetry that describes non-continuous transformations of a system. In

addition to continuous Lorentz transformation, there are two other space-time transformations that can

be symmetries of the Lagrangian: parity and time reversal. Parity, which is denoted by P , sends (t; ~x)!(t;�~x), while the time reversal, which is denoted by T , sends (t; ~x)! (�t; ~x). At the same time when we

discuss P and T , it will be convenient to discuss the discrete transformation: charge conjugation, which

is denoted by C. Under this transformation, the particles and antiparticles are interchanged. Although

any relativistic �eld theory must be invariant under the Poincaré group, it need not be invariant under

the discrete transformations P , T and C. From experiments, we know that the three of the four forces

of Nature, the gravitational, electromagnetic, and strong interactions, are symmetric under P , T , and C.

Page 8 of 193


The fourth force, the weak interactions, violates both P and C, and certain rare processes in the Yukawa

sector (processes involving neutral K mesons) also show CP and T violation. All experiments indicate

that CPT is a symmetry of Nature.

2.1.1 Realization of symmetries

So far, we have considered only exact symmetries. It is important to di�erentiate what actually is

symmetric, Lagrangian or the vacuum, and at what scale the symmetry is manifest, and if it is broken,

how it is broken. There is di�erent ways the symmetry can be broken. If the Lagrangian is invariant

under a symmetry for which the vacuum is not invariant this symmetry is termed spontaneously broken.

The symmetry can also be explicitly broken via adding non-invariant terms in the Lagrangian. It can

not be excluded that the symmetry cannot be used to draw conclusions, if the breaking term is small.

Some classical symmetries of the Lagrangian can be spoiled by the quantum e�ects, when we quantize

the theory. This is called an anomalous symmetry, and the term that gives the breaking is called an

anomaly. It is important for the consistency of the theory that all the local anomalies are cancelled in

the end, for example the gauge anomalies in the Standard Model is cancelled as shown in Appendix B.

The consequence by breaking symmetries is described by the Goldstone theorem (derived at quantum

level in Appendix D). The Goldstone theorem states the following: If a subgroup H of the symmetry

group G is broken, then there are dim(G=H) Goldstone bosons.

2.2 Standard Model

In this section, we schematic summarize the Standard Model (SM) and brie�y discuss each part of

its total Lagrangian. The SM is a SU(3)C SU(2)W U(1)Y gauge group. The three factors of the

gauge symmetry give rise to three fundamental interactions (electromagnetic, weak nuclear and strong

nuclear interactions). The SM has been hugely successful in explaining experimental observations, but

it leaves some phenomena unexplained. The SM does not incorporate the full theory of gravitation as

described classically by the general relativity, dark matter, dark energy, neutrino masses and oscillations

and baryogenesis. Therefore, the SM is not a complete theory of the fundamental interactions.

The theory of the strong nuclear interactions is a non-abelian gauge theory with the gauge group

SU(3)C. The quantum �eld theory (QFT) of these interactions is called the quantum chromodynamics

(QCD). This theory describes the interactions between quarks and gluons, which makes up hadrons

(mesons and baryons) such as protons, neutrons and pions. The force carriers (the gauge bosons) in the

theory are the gluons, and the associate charge is called color (see Table 2.2). The generators of QCD

are the eight Gell-Mann matrices �a.

The theory of the electroweak interaction is also a non-abelian theory with the gauge group SU(2)WU(1)Y. This gauge group is not simple, but it is a product of SU(2)W and U(1)Y, the uni�cation of

the electromagnetic with the charged and neutral weak interactions, which is the combination by two

Page 9 of 193


gauge coupling constants, g for the weak isospin SU(2)W and g0 for the weak hypercharge U(1)Y. The

force carriers of the non-abelian gauge group SU(2)W are the massless W a� bosons with a = 1; 2; 3

and of the abelian gauge group U(1)Y is the massless B� boson. The masses of the gauge bosons are

introduced by spontaneous symmetry breaking, when the Higgs boson, � = v + h, requires a non-zero

vacuum expectation value (vev), h�i = v, which provides three massive bosons called W� and Z and one

massless photon . This vev is also responsible for the fermion masses in the SM from the interactions

between fermions and the Higgs boson, which are described in the Lagrangian by the Yukawa terms. We

have three generators of weak isospin called IaW = �a=2 (�a are the Pauli matrices) and the generator

of weak hypercharge YW. The EW gauge group spontaneously breaks to the electromagnetic symmetry

group, i.e. SU(2)W U(1)Y ! U(1)Q, when the Higgs requires a vev. The QFT of this gauge theory

is called quantum electrodynamics (QED). The generator of the electric charge is de�ned via the Gell-

Mann-Nishijima relation (cf. Eq. (4.2.1) in Ref. [13])

Q = I3W +YW

2: (2.8)

The electrical charges of the various particles in the SM are shown in Table 2.2.

SCALARS

Symbol Name Electric charge Baryon number Lepton number Gauge representations

� Higgs doublet (1,0) 0 0 (1,2,1)

FERMIONS

Symbol Name Electric charge Baryon number Lepton number Representation

QLI Left-handed quark (2/3,-1/3) 1/3 0 (3,2,1/3)

uRI Right-handed up quark 2/3 1/3 0 (3,1,4/3)

dRI Right-handed down quark -1/3 1/3 0 (3,1,-2/3)

LLI Left-handed lepton (0,-1) 0 1 (1,2,-1)

eRI Right-handed electron lepton -1 0 1 (1,1,2)

GAUGE FIELDS

Symbol Associate charge Electric charge Group Coupling Gauge Gauge representations

B Weak hypercharge 0 U(1)Y g' (1,1,0)

W 1;2;3 Weak isospin 0 SU(2)W g (1,3,0)

G Color 0 SU(3)C gs (8,1,0)

Table 2.2: The content of �elds in the SM. If we have a doublet (e.g. the Higgs doublet �) then its eachelectric charges are represented as (U(1)Q charge of �rst component, U(1)Q charge of second component),e.g. (1,0) for the Higgs doublet. The representations of the �elds under the gauge groups SU(3)C, SU(2)Wand U(1)Y are listed as (SU(3)C, SU(2)W, U(1)Y). For example, the gluons have the gauge representations(8;1; 0), because there is a color octet of gluons, which all are weak isospin singlets with hyperchargezero.

Page 10 of 193


It is often convenient to denote the left-handed quarks and leptons doublets by

QLI =

0@ uLI

dLI

1A and LLI =

0@ �LI

eLI

1A ; (2.9)

and the right-handed fermion singlets by uRI , dRI , and e

RI . Here u; d; �, and e represent up-type quark,

down-type quark, neutrino, and electron-type lepton, respectively, and I is the generation index (I; J;K; � � � =1; 2; 3). The three di�erent generations of the SM contain

uI =fu; c; tg; dI = fd; s; bg; �I = f�e; ��; ��g; eI = fe; �; �g: (2.10)

The content of the �elds in the SM and their quantum numbers is shown in Table 2.2. The representations

of the �elds under the gauge groups SU(3)C, SU(2)W and U(1)Y are listed as (SU(3)C, SU(2)W, U(1)Y) in

the table. For example, the gluons form a color octet which all are weak isospin singlets with hypercharge

zero, i.e. they have the gauge representations (8,1,0).

Finally, we will formulate the Lagrangian of the SM. The Lagrangian of the SM must respect the

gauge symmetries, Lorentz invariance and renormalization. It is useful to divide the total Lagrangian

into four parts as follows

LSM = LG + LF + LH + LY: (2.11)

The �rst term contains the Yang-Mills terms for the gauge �elds, which reads

LG = �1

4W i��W

i�� 1

4B��B

�� 1

4Ga��G

a�� ; (2.12)

where the gauge �eld strength tensors are de�ned as

W i�� =@�W

i� � @�W i

� + g"ijkW j�W

k� ;

B�� =@�B� � @�B�;Ga�� =@�G

a� � @�Ga� + gsf

abcGb�Gc� ;

(2.13)

where i = 1; 2; 3 and a = 1; : : : ; 8. The structure constants are de�ned as [�a; �b] = ifabc�c and [�i; �j ] =

i"ijk�k, where �a and �i are the generators of SU(3)C and SU(2)W gauge group, respectively. The second

term is the fermion terms, the kinetic term and their interactions with the gauge bosons, which are

LF =XI

(�LLI i =DLLI + �QLI i =DQ

LI ) +

XI

(�eRI i =DeRI + �uRI i =Du

RI + �dRI i =Dd

RI ); (2.14)

where the covariant derivative is

D� = @� � ig �i

2W i� + ig0

YW

2B� � igs�

a

2Ga�: (2.15)

The third part of the Lagrangian contains only the Higgs and the electroweak gauge bosons

LH =(D��)y(D��)� V (�)

=(D��)y(D��) + �2�y�� (�y�)2;

(2.16)

Page 11 of 193



D� = @� � ig �i

2W i� + ig0

YW2B�; (2.17)

and the Higgs complex doublet is

�(x) =

0@ �+(x)

�0(x)

1A =

1p2

0@ �2(x) + i�1(x)

�(x)� i�3(x)

1A : (2.18)

We have that �� = (�2 � i�1)=p2. The mass terms for the gauge bosons come from the kinetic term

after the Higgs boson has acquired a vev, which is v = 246 GeV. Therefore, the physical Higgs �eld, h, is

an excitation around the vev, v, and we would write � = v + h with the expectation value h�i = v. The

physical content of EW symmetry breaking can be extracted most easily in the unitary gauge, where the

would-be Goldstone boson components, �1;2;3, are set to zero (cf. page 582 in Ref. [13]). In this gauge,

there are no unphysical �elds and we can classify the physical �elds as eigenstates of electric charge and

mass. In this gauge, the Higgs doublet is thus

�(x) =1p2

0@ 0

v + h

1A : (2.19)

A mass term of the form �m2W a�W

a� for the gauge bosons is not invariant under non-Abelian SU(2)W

gauge transformations in Eq. (2.7), and therefore it is forbidden. The spontaneous electroweak (EW)

symmetry breaking of the following terms in the kinetic term of Eq. (2.16) gives

g2

4�y�W a

�Wa� +

g02

4�y�B�B� +

1

4�y�(gW 3

� + g0B�)(gW 3� + g0B�) SB��!g2

8W a�W

a�(v + h)2 +g02

8B�B

�(v + h)2 +1

8(gW 3� + g0B�)2(v + h)2 =

g2v2

4W+� W

�� +(g2 + g02)v2

4Z�Z

� ++g2v

2hW+

� W�� +

(g2 + g02)v2

hZ�Z�+

g2

4hhW+

� W�� +

g2 + g02

8hhZ�Z

�:

(2.20)

We have that the mass eigenstates of the gauge bosons are

W�� (x) =

1p2

�W 1�(x)� iW 2

�(x)�

and0@ A�

Z�

1A =

0@ cW sW

�sW cW

1A0@ B�

W 3�

1A (2.21)

with the weak mixing angle (the Weinberg angle)

cW � cos �W =gp

g2 + g02and sW � sin �W =

g0pg2 + g02

; (2.22)

which rotates the originalW 3 and B vector boson plane. This rotation gives one positively and negatively

charged gauge boson, W�� bosons, two neutral gauge boson, Z� boson and the photon A�. According to

Eq. (2.20), the masses at tree level of these gauge bosons are

Page 12 of 193


mW =gv

2; mZ =

pg2 + g02v

2=mW

cW; m = 0: (2.23)

The Higgs couplings to the massive gauge bosons are

gSMhWW =g2v

2; gSMhZZ =

(g2 + g02)v2

;

gSMhhWW =g2

4; gSMhZZ =

g2 + g02

8:

(2.24)

In the future, we would de�ne I as a Dirac spinor in the generation I; J;K; � � � = 1; 2; 3, which can be

either uI , dI , �I or eI . The left- and the right-handed can be projected out with the projection operators

PL;R = (1 � 5)=2 as follows L;R = PL;R . So far all the fermions are massless. A Dirac mass term is

not allowed, because the SU(2)L symmetry transforms the �eld eL into another �eld �L. Under such a

transformation the mass term (cf. Eq. (7.167) in Appendix C-2)

�m � = �m( � L R + � R L) (2.25)

is clearly not invariant, and therefore it is forbidden. Again, we can generate a mass term via the Higgs

mechanism. We can construct a term that is a product of the Higgs and one of the SU(2)L doublets

of the left-handed fermions as in Eq. (2.9). These terms are called Yukawa interaction terms, and the

Yukawa Lagrangian in the SM is

LY =� �Q0LI GuIJu

0RJ �c � �Q0LI G

dIJd

0RJ �� L0LI G

eIJe

0RJ �+ h.c. (2.26)

where Ge, Gu and Gd are 3 � 3 matrices, and the fermion �elds 0L;RI are the charge eigenstates of the

weak interaction. The �eld �c(x) is the charge-conjugate Higgs �eld �c(x) = i�2��(x) = (�0�;��(x))(cf. page 595 in Ref. [13]), where �a are the Pauli matrices in Eq. (7.3) in Appendix A. It follows that

the conjugated Higgs �eld �c(x) also transforms as a SU(2) doublet, because the identity

i�2 exp(��a�a�=2) = exp(i�a�a=2)i�2: (2.27)

After the spontaneous symmetry breaking (� = v + h), we have the terms

�G IJ(� � L;I R;J + h.c.)SB��! � vp

2G IJ(

� L;I R;J + h.c.) = �M IJ(

� L;I R;J + h.c.); (2.28)

where the mass matrices for up-type quarks, down-type quarks, and electron-type leptons are

MuIJ =

vp2GuIJ ; Md

IJ =vp2GdIJ ; and Me

IJ =vp2GeIJ : (2.29)

These mass matrices can be diagnoalized by a bi-unitary transformation for left-handed and right-handed

fermions, respectively, resulting in the fermion mass eigenstates,

0LI =XK

U ;LIK LK and 0RI =XK

U ;RIK RK : (2.30)

Thus, the fermion masses are

Page 13 of 193


m ;I =1p2

XK;M

U ;LIK G KMU ;RyMI v: (2.31)

Thus, the fermions are no longer in the charge eigenstates of weak interaction, but they are mass eigen-

states. Thus, the �rst term in the Yukawa Lagrangian in Eq. (2.26) can be written as

� �Q0LI GuIJu

0RJ �c =�

XI

mu;I

2v�uI(1 + 5)uI(v + h� i�3) +

XI;J

�dLI VqyIJ

p2mu;J

vuRJ �

�; (2.32)

and the second term is

� �Q0LI GdIJd

0RJ � =�

XI;J

�uLI VqIJ

p2mu;J

vdRJ �

+ +XI

md;I

2v�dI(1 + 5)dI(v + h+ i�3); (2.33)

and the third term is

��L0LI GeIJe

0RJ � = �

XI;J

��LI VlIJ

p2me;J

veRJ �

+ +XI

me;I

2v�eI(1 + 5)eI(v + h+ i�3); (2.34)

which are derived in Eqs. (7.168)-(7.170) in Appendix C-2. We have used Eq. (2.30), Eq. (2.31) and the

anticommutation relation f 5 �g = 0 to rewrite these Yukawa terms. The neutral currents which are

not changing �avors, the combinations U ;L(U ;L)y = 1 always appear, and they are not a�ected. For

the �avor-changing currents we have the matrices

V q =Uu;L(Ud;L)y;

V l =U�;L(Ue;L)y;(2.35)

which providing the �avor mixing. The matrix V q is the CKM matrix for quark mixing, and the matrix

V l is the PMNS matrix for possible lepton mixing.

By inserting the Yukawa terms in Eqs. (2.32)-(2.34) into Eq. (2.26), we obtain that the total Yukawa

Lagrangian in the SM can be written in terms of Dirac spinors as follows

LY =� �Q0LI GuIJu

0RJ �c � �Q0LI G

dIJd

0RJ �� L0LI G

eIJe

0RJ �+ h.c.

=�X

f=u;d;e

XI

mf�fIfI �

Xf=u;d;e

XI

mf;I

v( �fIfIh� 2I3w;f i

�fI 5fI�3)

+XI;J

p2

v

�mu;I

��uRI V

qIJd

LJ�

+ + �dLI VqyIJ u

RJ �

��md;J

��uLI V

qIJd

RJ �

+ + �dRI VqyIJ u

LJ�

��

�XI;J

p2

vme;J

��LI V

lIJe

RJ �

+ + �eRI VlyIJ�

LJ �

��;

(2.36)

where we have used Eq. (7.171) in Appendix C-2. Thus, the four Lagrangian parts in the total Lagrangian

of the SM in Eq. (2.11) are given in Eq. (2.12), Eq. (2.14), Eq. (2.16) and Eq. (2.36), respectively.

In the following section, we will investigate how the Higgs boson unitarizes the SM scattering ampli-

tudes. We will examine the unitarity of WLWL scattering amplitude. In this discussion we will discover

that we need a scalar particle to unitarize this scattering process, because the scattering amplitude grows

with the energy s=m2W without the scalar, where s is the center-of-mass (CM) energy squared of the

Page 14 of 193


WLWL scattering. As we will see, the current data on the observed Higgs coupling to W bosons still

allow room for additional doublets besides the discovered Higgs boson. Therefore, it remains an open

question, whether the discovered Higgs boson is the only one responsible for the full EW symmetry

breaking.

2.3 Unitarity of WLWL Scattering Amplitude

A new scalar particle with mass of approximately 125 GeV was discovered at the Large Hadron Collider

(LHC) in July 2012. This is consistent with being the long sought Higgs boson of the SM, which

was proposed in 1960s (in Refs. [59�61]). This Higgs particle cures would-be violation of unitarity of

the scattering amplitudes in the SM. In this section, we will show that scattering of the longitudinal

components of the weak gauge bosons is a useful probe of EW symmetry breaking. The SM scattering

amplitudes (e.g. the amplitudes of the diagrams in Figure 2.1) in the SM without Higgs exchange grow

with the energy as s=m2W , where s is the center-of-mass (CM) energy squared of the WLWL scattering.

I.e. the amplitude of the WLWL scattering diverges with the energy, and thus it is not unitary. Including

the Higgs boson exchanges as shown in Figure 2.2, total WLWL scattering amplitude is unitarized.

W+L;�(p1)

W�L;�(p2)

W+L;�(q1)

W�L;�(q2)

W+L;�(p1)

W�L;�(p2)

W+L;�(q1)

W�L;�(q2)

W+L;�(q1)

W�L;�(q2)

W+L;�(p1)

W�L;�(p2)

Z; Z;

p1 + p2

p1 � q1

Figure 2.1: The diagrams that contribute to the amplitude of the WLWL scattering with purely weakgauge bosons contributions.

W+L;�(p1)

W�L;�(p2)

W+L;�(q1)

W�L;�(q2)

W+L;�(q1)

W�L;�(q2)

W+L;�(p1)

W�L;�(p2)

hh

p1 + p2

p1 � q1

Figure 2.2: The diagrams that contribute to the amplitude of theWLWL scattering with the Higgs bosoncontributions.

Now, we consider the process W+(p1)W�(p2)! W+(q1)W

�(q2), which gets contributions from the

Feynman diagrams of a four-point vertex and ; Z in both s and t channels, as well as the diagrams with

a Higgs propagator in both s and t channels. The amplitudes for the gauge diagrams in Figure 2.2 can

be written as

Page 15 of 193


iM4(WL;WL !WL;WL) =ie2

s2W[2�L(p2) � �L(q1)�L(p1) � �L(q2)� �L(p2) � �L(p1)�L(q1) � �L(q2)

� �L(p2) � �L(q2)�L(p1) � �L(q1)]

iMZ; s (WL;WL !WL;WL) =� ie2

�1

s+c2w=s

2W

s�m2Z

� h(p1 � p2)��L(p1) � �L(p2) + 2p2 � �L(p1)��L(p2)

� 2p1 � �L(p2)��L(p1)ih(q2 � q1)��L(q1) � �L(q2)� 2q2 � �L(q1)�L;�(q2)

+ 2q1 � �L(q2)�L;�(q1)i

iMZ; t (WL;WL !WL;WL) =� ie2

�1

t+c2w=s

2W

t�m2Z

� h(p1 + q1)

��L(p1) � �L(q1)� 2q1 � �L(p1)��L(q1)

� 2p1 � �L(q1)��L(p1)ih(p2 + q2)��L(p2) � �L(q2)� 2q2 � �L(p2)�L;�(q2)

� 2p2 � �L(q2)�L;�(p2)i;

where ��L(k) is the longitudinal polarization four-vectors of the W bosons with momentum k. The

amplitudes for the Higgs boson diagram in Figure 2.1 are given by

iMHiggss (WL;WL !WL;WL) =�

�L(p1)i

emW

sWg��

�L(p2)

i

(p1 + p2)2 �m2h

��L(q1)iemW

sWg��

�L(q2)

iMHiggst (WL;WL !WL;WL) =�

�L(q1)i

emW

sWg��

�L(p1)

i

(p1 � q1)2 �m2h

��L(q2)iemW

sWg��

�L(p2):

(2.37)

To calculating these amplitudes we need the longitudinal polarization four-vectors ��L, which are de-

rived in Appendix C-2 in Eqs. (7.175)-(7.184). In the center-of-mass frame of the incomingW+(p1)W�(p2)

pair where ~p1 = �~p2, according to Eq. (7.183) and Eq. (7.184) we can express the longitudinal polariza-

tion four-vector as

��L(p1) =p�1mW

� 2mW

sp�2 ; (2.38)

and similarly

��L(p2) =p�2mW

� 2mW

sp�1 ; (2.39)

where s = (p1 + p2)2 = 4 (p0)

2. For the outgoing W+

L (q1)W�L (q2) pair their longitudinal polarization

vectors can be obtained by simply make the substitution (p1; p2) ! (q1; q2). However, we need to write

the various products between the four-momentum vectors in terms of the Mandelstam variables, s, t and

u, in Eq. (7.172) as in Appendix C-2, which are given in Eq. (7.173). Finally, we need also the relation

where the sum of Mandelstam variables gives s+ t+ u = 4m2W.

These longitudinal polarization vectors and these expressions can be substituted into the above am-

plitudes, to leading term of order O(E4=m4W ) of each amplitude we have calculated them in Eqs. (7.185)-

(7.195) in Appendix C-2 to be

M4(WL;WL !WL;WL) =e2

4m4W s

2W

�s2 + 4st+ t2 � 4m2

W (s+ t)� 8m2W

sut

�+O

��EmW

�0�;

MZ; s (WL;WL !WL;WL) = � e2

4m4W s

2W

�s(t� u)� 3m2

W (t� u)�+O

��EmW

�0�;

Page 16 of 193


MZ; t (WL;WL !WL;WL) = � e2

4m4W s

2W

�t(s� u)� 3m2

W (s� u) + 8m2W

su2�+O

��EmW

�0�;

MHiggss (WL;WL !WL;WL) = � e2

4s2Wm2W

�s� 2m2

W

�2s�m2

h

+O��

EmW

�0�; (2.40)

MHiggst (WL;WL !WL;WL) = � e2

4s2Wm2W

�t� 2m2

W

�2t�m2

h

+O��

EmW

�0�:

The sum of the gauge diagrams in Figure 2.1 (cf. Eq. (7.174 in Appendix C-2) is

MGauge (WL;WL !WL;WL) =M4 +MZ; s +MZ;

t = � e2

4s2Wm4W

u+O��

EmW

�0�: (2.41)

The gauge structure ensures the cancellation of the O(E4=m4W ) terms. The problem is that the sum

of the gauge diagrams are left with O(E2=m2W ). Therefore, for the scattering amplitudes with purely

gauge bosons without Higgs bosons, the amplitudes grow with the energy as s=m2W ,1 and thus it is not

unitarized.

However, we have the contributions from the Higgs diagrams in Figure 2.2, which are

MHiggs (WL;WL !WL;WL)

=MHiggss +MHiggs

t = � e2

4s2Wm2W

"�s� 2m2

W

�2s�m2

h

+

�t� 2m2

W

�2t�m2

h

#+O

��EmW

�0�

' � e2

4s2Wm2W

(s+ t) +O��

EmW

�0�= � e2

4s2Wm2W

�4m2

W � u�+O�� EmW

�0�

=e2

4s2Wm2W

u+O��

EmW

�0�(2.42)

in the limit s� m2h;m

2W . Totally, the WLWL scattering amplitude is

MTotal =MHiggs +MGauge = O��

EmW

�0�: (2.43)

Therefore, in the SM with the Higgs boson the amplitude is completely unitarized by the Higgs

boson. Onceps goes beyond the Higgs boson mass, then the scattering amplitude will no longer grow

like s=m2W . However, because current data still contains signi�cant uncertainties. There is still room

for a non-SM Higgs sector, e.g continaing more doublets like in the 2HDM. One important measurment

is the constraints of the Higgs coupling to W and Z bosons pair in Eq. (2.24), i.e. gSMhWW = g2v=2

and gSMhZZ = (g2 + g02)v=2, respectively. The current data for the ratio of this coupling measured for the

combination of the ATLAS and CMS measurements and in the SM (cf. table 18 in Ref. [36]) is

�W � ghWW

gSMhWW

= 0:91+0:10�0:12 (2.44)

with 1� CL intervals. The central value is close to one, i.e. that the observed Higgs boson leaves little

space for the existence of another Higgs boson or some physics beyond the SM which couples to the W�

bosons.

1The Mandelstam variable s is related to the other two Mandelstam variables, e.g. u, with the relation s+ t+u = 4m2

W.

Page 17 of 193


On the other hand, if the Higgs boson coupling to the W bosons deviates from the SM value, then

the amplitude for the Higgs diagrams in Figure 2.2 is modi�ed with the ratio �W in Eq. (2.44) squared

from the two hWW vertices in the diagrams. The sum of the modi�ed Higgs amplitudes are

MHiggs (WL;WL !WL;WL) = �2We2

4s2Wm2W

u+O��

EmW

�0�: (2.45)

Even for small deviation from the SM value, the terms grow like u=m2W (related to s=m2

W) in the scattering

amplitude, and these terms would blow up after hitting the mass pole of the Higgs boson. Thus, according

to Eq. (2.44) there is a possibility for a new Higgs doublet or other physics beyond the SM which is

responsible for unitarity of the scattering amplitude. The importance of this section is that we need a

scalar or something else to unitarize theWLWL scattering amplitude, where the Higgs boson is responsible

for it in the �rst place.

The next important feature discussed in next section is the custodial symmetry. In the SM, there

is a global symmetry of the Higgs potential in the SM (in Eq. (2.16)). In this section, we will give the

constraints on the size of the break of the custodial symmetry measured by the LEP experiments, which

gives only small room for models beyond the SM that breaks the custodial symmetry.

2.4 Custodial Symmetry

One aspect we have glossed over so far is the necessity of two doublets for the Higgs. When we have

only one doublet, then in this case the number of Goldstone bosons is too small to provide three massive

gauge bosons which is demanded by experiment. Thus, the presence of two doublets is experimentally

necessary. The extra doublet has more consequences than just giving all three weak gauge bosons mass.

2.4.1 Custodial Symmetry at Tree Level

To understand these consequences it is best to concentrate �rstly on the pure Higgs part of the Lagrangian,

LH, in Eq. (2.16), which reads

LH = (D��)y(D��)� V (�) = (D��)

y(D��)� �2�y�� (�y�)2; (2.46)

where the Higgs doublet is

�(x) =1p2

0@ �2(x) + i�1(x)

�(x)� i�3(x)

1A : (2.47)

Rewriting the Higgs doublet in Eq. (2.47) with four degrees of freedom to a 2� 2 matrix

M(x) � 1p2(�(x) + i�i�i(x)) = (�c(x);�(x)) =

1p2

0@ �(x) + i�3(x) �2(x) + i�1(x)

��2(x) + i�1(x) �(x)� i�3(x)

1A ; (2.48)

Now, the pure Higgs Lagrangian in Eq. (2.46) can be rewritten to

LH =1

2Tr[D�M(D�M)y]� �2

2Tr[MMy]� �

4Tr[MMy]2; (2.49)

Page 18 of 193



D�M = @�M � igW a�

�a

2M + ig0MB�

�3

2: (2.50)

This Lagrangian is identical to the Lagrangian in Eq. (2.46). This can be seen by multiplying two

doublets together, which gives

�y� =1

2

��2 + �21 + �22 + �23

�and

1

2Tr[MMy] =

1

2

��2 + �21 + �22 + �23

�;

(2.51)

and by writing out for example the �rst term without derivative in the kinetic term, which gives

��ig �12 W 1

��ig �12 W 1��

�y=g2

8[�2 + �21 + �22 + �23 ]W

1�W

1� and

1

2Tr

��igW 1

��1

2 M��igW 1� �1

2 M�y�

=g2

8[�2 + �21 + �22 + �23 ]W

1�W

1�:

(2.52)

This can also be shown for the rest of the terms in the kinetic term. By inserting these expressions above

into Eq. (2.49), we obtain the Higgs Lagrangian in Eq. (2.46) again.

The next thing, we do, is to set the EW coupling constants g; g0 = 0, then we have the global symmetry

group SU(2)L SU(2)R, which is isomorphic to SO(4) (i.e. SU(2)L SU(2)R �= SO(4)). By rewriting

the Lagrangian with the doublet � to one with the M matrix, we can now see the SU(2)L and SU(2)R

symmetries. We have namely that the Lagrangian with the M matrix in Eq. (2.49) is invariant under

the global transformation

M ! gLMgyR; (2.53)

where gL 2 SU(2)L and gR 2 SU(2)R, because

Tr[MMy]! Tr[gLMgyRgRMygyL] = Tr[gLMMygyL] = Tr[MMy]; (2.54)

and therefore the kinetic term is also invariant under these global transformations. Thus, we have

rewritten the Higgs Lagrangian in a form such that we can see both the SU(2)L and the SU(2)R symmetry.

When we take the mass parameter �2 to be negative and the self-coupling � positive, then at tree level

we obtain

h�i2 � v2 =j�2j�

6= 0 and � = v + h; (2.55)

where h is the Higgs �eld and v is vev of the Higgs �eld. The global symmetry group SO(4) will break to

SO(3) which is isomorphic to SU(2)V, i.e. that SU(2)L SU(2)R ! SU(2)V, when one of the degrees of

freedom gets �xed, because the expectation value in one direction is di�erent from zero, h�y�i = v2=2.

The symmetry group SU(2)L SU(2)R breaks to SU(2)V, because

h�i = hTr(M)i ! Tr(gLMgyR)

�= hTr(gyRgLM)i = hTr(M)i; (2.56)

if and only if gL = gR = gV . I.e. that the global group breaks to SU(2)V, when the Higgs has a vacuum

expectation value.

Page 19 of 193


Now, we are gauging the weak isospin SU(2)L and the hypercharge U(1)Y gauge group by setting

the coupling constants g 6= 0 and g0 6= 0 in the covariant derivative in Eq. (2.50). More precisely the

SU(2)R symmetry is broken explicitly, because the U(1)Y subgroup of it is gauged. This can be seen in

the following term in the kinetic term of the Lagrangian in Eq. (2.49) by transforming it,

1

2Trh��igW a

��a

2 M� �ig0MB� �

3

2

�i= �gg

0

8W a�B

�Tr[�aM�3My]!

� gg0

8W a�B

�Tr[�agLMgyR�3gRM

ygyL] 6= �gg0

8W a�B

�Tr[�aM�3My];(2.57)

which is not invariant under the global transformation M ! gLMgyR of the symmetry group SU(2)L SU(2)R. Therefore, the hypercharge gauge group U(1)Y breaks the global symmetries down to a subgroup

SU(2)W U(1)Y, when it is gauged (i.e. when g0 6= 0). Thus, we have that the M matrix transforms

now globally as M ! gWMgyY to keep the Lagrangian invariant, where gW 2 SU(2)W and gY 2 U(1)Y.

Overall, when we are gauging some of the symmetry groups, then we break some of the global symmetries.

Therefore, we have now the following symmetry breaking pattern for the EW symmetry breaking

SU(2)W U(1)Y ! U(1)Q; (2.58)

where U(1)Q is the electromagnetic gauge symmetry. This gives the three massless Goldstone bosons,

i.e. �� (1=p2)(�1� i�2) and �0 � �3, according to the Nambu-Goldstone's theorem. These Goldstone

bosons become the longitudinal degree of freedom of the massive electroweak gauge bosons W� and Z

in the unitary gauge.

If the coupling constants are g 6= 0 and g0 = 0, then we have still the symmetry breaking pattern

SU(2)L SU(2)R ! SU(2)V. In this case, the covariant derivative is

D�M = @�M � igW a�

�a

2M: (2.59)

By inserting � = v + h into the kinetic term in the Lagrangian in Eq. (2.49), we obtain the mass term

of the W i bosons

Tr[D�M(D�M)y] =Trh�@�M � igW a

�

�a

2M��@�M

y + igW a�M

y�a

2

�i=Tr

hg2W a

��a

2 MMy �b2 W

b�i+ � � � = Tr

hg2

4 Wa��

a v2

2 12�2�bW b�

i+ : : :

=Trhg2v2

8 W a� �

ab12�2W b�

i+ Tr

hg2v2

8 W a� i"

abc�cW b�i+ : : :

=g2v2

4W a�W

a� + : : : :

(2.60)

According to Eq. (2.21) we have for g0 = 0 that

W+ =W 1 � iW 2

p2

; W� =W 1 + iW 2

p2

and 00Z 00 =W 3; (2.61)

and therefore we get that

W+� W

�� =1

2(W 1

�W1� +W 2

�W2�) and 00Z�Z�00 =W 3

�W3�: (2.62)

Page 20 of 193


By inserting these expressions into Eq. (2.60), we obtain the mass terms of W�� and 00Z 00

Tr[D�M(D�M)y] =g2v2

4W a�W

a� + � � � = g2v2

4(W 1

�W1� +W 2

�W2� +W 3

�W3�) + : : :

=g2v2

4(W+

� W�� + 00Z�Z�00) + : : : ;

(2.63)

i.e. that the masses of the W� and 00Z 00 bosons are degenerated, which are mW = mZ = gv=2. This

degeneration is because of the custodial symmetry which is the global symmetry group SU(2)V. In this

case, the Weinberg angle in Eq. (2.22) is

cos �W =gp

g2 + g02= 1: (2.64)

Therefore, we have the following relation between the masses and the Weinberg angle

mZ

mW cos �W= 1: (2.65)

On the other hand, if both coupling constants are di�erent from zero, g 6= 0 and g0 6= 0, then we have

the symmetry breaking pattern SU(2)W U(1)Y ! U(1)Q. In that case, the covariant derivative of M is

D�M = @�M � igW a�

�a

2M + ig0M

�3

2B�: (2.66)

When the Higgs �eld requires a vacuum expectation value h�i = v, then we obtain the following terms

from the kinetic term

Tr[D�M(D�M)y] =Trhg2W a

��a

2 MMy �b2 W

b�i+ Tr

hg02M �3

2 B��3

2 MyB�

i�

Trhgg0W 3

��3

2 M�3

2 MyB�

i� Tr

hgg0M �3

2 B�W3�M

y �32

i=g2v2

4W a�W

a� +g02v2

4B�B

� � 2gg0v2

4W 3�B

� + : : :

=v2

4(g2W+

� W�� + g2W 3

�W3� + g02B�B� � 2gg0W 3

�B�) + : : : :

(2.67)

According to Eq. (2.21), we have

Z�Z� =

1

g2 + g02(�g0B� + gW 3

�)(�g0B� + gW 3�)

=1

g2 + g02(g02B�B� + g2W 3

�W3� � 2gg0W 3

�B�):

(2.68)

Therefore, we get that the mass terms for the W�� and Z� bosons are

Tr[D�M(D�M)y] =g2v2

4W+� W

�� +v2

4(g2 + g02)Z�Z�: (2.69)

The custodial symmetry is now broken, and therefore the masses of the gauge bosons are no longer

degenerate:

mW =gv

2and mZ =

v

2

pg2 + g02: (2.70)

The relation between the masses of the W bosons and the Z boson at tree level can be written as

Page 21 of 193


1 =mW

cos �WmZ

=gv=2

cos �w(v=2)pg2 + g02

=g

cos �Wpg2 + g02

)

cos �W =gp

g2 + g02;

(2.71)

Thus, the custodial symmetry is hidden in the cosine of the Weinberg angle cos �W. We will see in the

following that the Yukawa sector breaks the custodial symmetry.

2.4.2 Custodial Symmetry at Loop Level

The Yukawa interactions do not respect the custodial symmetry. At loop level, there is a very small

additional contribution to the left-hand side of the relation in Eq. (2.71). We de�ne that the right-hand

side is equal to some � parameter, such that we obtain the mass relation

m2W

m2Z cos

2 �W� � � 1 + ��: (2.72)

The Lagrangian with the Yukawa interactions for the quarks in Eq. (2.36) without �avor mixing is

rewritten in terms of Weyl spinors, which for one generation has the form

LY =� �u"ijqLj�iu�R � �dqyLi�idR + h.c.

=� �u"ijqLj�iu�R � �dqyLi�idR � �u"ijqyLi�yjuR � �dqLi�yid�R;(2.73)

where the Yukawa couplings are �q =p2mq=v and i; j; � � � = 1; 2 are SU(2)W indices. By using the M

matrix in Eq. (2.48) the Yukawa terms can be rewritten to

� qyLM0@ uR

dR

1A+ h.c.; (2.74)

which is invariant under the custodial symmetry

� qyLM0@ uR

dR

1A+ h.c.! �qyLgyLgLMgyRgR

0@ uR

dR

1A+ h.c. = �qyLM

0@ uR

dR

1A+ h.c.: (2.75)

The problem is that this is for the case where the Yukawa coupling constants, �u and �d, are the same,

but this is not the case in the SM. The Yukawa terms can instead be rewritten to

LY = ��uqyLMPU

0@ uR

dR

1A� �dqyLMPD

0@ uR

dR

1A+ h.c.; (2.76)

where

PU =

0@ 1 0

0 0

1A and PD =

0@ 0 0

0 1

1A : (2.77)

These terms are not invariant under custodial transformations,

LY !� �uqyLgyLgLMgyRPUgR

0@ uR

dR

1A� �dqyLgyLgLMgyRPDgR

0@ uR

dR

1A+ h.c. =

Page 22 of 193


� �uqyLMgyRPUgR

0@ uR

dR

1A� �dqyLMgyRPDgR

0@ uR

dR

1A+ h.c. 6= LY ;

because PU or PD do not commute with gR. Therefore, the Yukawa terms in the SM are not custodial

symmetric, if the masses of the fermions in each generation are di�erent.

It will give rise to contributions to the � parameter from the loop diagrams, which lead to the mass

corrections to the masses of the W� and Z bosons. In Figure 2.3 are shown the one-loop diagrams that

lead to mass corrections of the two masses with the two heaviest fermions, the top and the bottom quark.

The Yukawa coupling is expressed in the fermion propagators in the loops. These kinds of corrections to

the � parameter (cf. Eq. (120) in Ref. [44]) are

��f =3�v2

16�m2Ws

2W

�mf

v

�2' 0:018

�mf

v

�2: (2.78)

where the electromagnetic �ne-structure constant �(mZ) = 127:950 � 0:017 (Ref. [44]), mW = 80:428 �0:039 GeV (Ref. [73]) and s2W = 0:2236 � 0:0041 (Ref. [73]).2 This correction is very small even for the

heaviest fermions with masses mt ' 172 GeV and mb ' 4 GeV compared to the vev of the Higgs boson

v = 246 GeV. This gives ��t = 0:0088 for the top-loop correction.

In Appendix I, the so-called T parameter is de�ned in terms of the self-energy of the vector bosons,

which is one of the EW precision parameters measured at the LEP experiments. The T parameter is

normalized to be zero in the SM. Data from the LEP experiments constrain the T parameter to be

T = 0:08 � 0:12 (cf. Eq. (10.72) in Ref. [73]). It is related to the � parameter (cf. Eq. (10.68) in

Ref. [73]) as follows

� =1

1� �(mZ)T' 1 + �(mZ)T: (2.79)

Therefore, the experimental measurement of the � parameter from the LEP experiments is � = 1:0006�0:0009, where the loop corrections in the SM are included such that the � parameter is normalized to be

one in the SM.

W� W�

t; b

Z Z

t; b

Figure 2.3: The loop diagrams which give rise to mass corrections of the W� and Z bosons and furtherprovide a correction to the � parameter.

If we consider an extra fermion doublet (U;D) with the usual left-handed coupling to SU(2)W, hy-

percharge Y and masses mU and mD, then it contributes to the T parameter (cf. Eq. (4.2) in Ref. [42])

2Parameters in QFTs depend on which energy scale they are measured, e.g. the mass �(mZ) is renormalized at the Zboson mass. This is discussed more clearly later, when we talks about the running of couplings.

Page 23 of 193


with

T � 1

12�s2Wc2W

(mU �mD)2

m2Z

� (mU �mD)2

v2; (2.80)

where we assume that jmU �mDj � mU;mD. Therefore, the increase of the T parameter by adding a

fermion doublet depends on the squared ratio of the mass splitting between the two doublet components

and the EW vev. Thus, the third generation of the quarks in the SM model contributes mostly to the T

parameter.

We can conclude, if the � parameter is measured to be greater than the SM's predictions of it, then

there should be something new physics. Experimentally, the � parameter is measured to be very close

to one. If we want to extend the SM, then the extension must only provide a very small contribution to

the � parameter. Therefore, new physics should be custodial symmetric or the symmetry must be broken

minimal.

In the following section, we will focus on the possible problems, called triviality and vacuum stability.

Triviality is a possible problem in QED and in a pure Higgs sector, while it turns out that possibly the

SM is vacuum unstable.

2.5 Triviality and Vacuum Stability

In (non-conformal) quantum �eld theory a change of the renormalization group (RG) scale � induces a

change in the coupling constants g of the theory. We say that the coupling constants run with energy.

The running of the coupling constants encodes important features of a theory, e.g. asymptotic freedom,

triviality, vacuum stability and uni�cation etc.. Let us examine two potential problems in the SM related

to the running couplings, which are triviality and the vacuum stability problem.

g

�

�

g

g

�

�

g

g

�

�

gE0

Asymptotic Theory

Trivial Theory

Unstable Theory

�L

Figure 2.4: Di�erent examples of how a coupling g can run and its �-function for a trivial, an asymptoticand an unstable theory.

Page 24 of 193


The dependence of g(�) upon � can be expressed in terms of a �-function of the theory (cf. Eq. (2.6.14)

in Ref. [13]),

�@g(�)

@�= �[g(�)]; (2.81)

where in Appendix E it is shown how the �-function is derived for di�erent coupling constants to one-loop

order.

A trivial theory means that the theory needs to be non-interacting (the coupling is zero) to be a

consistent theory. As we will see there is a Landau pole problem in QED, because its coupling constant

blows up to in�nity at a �nite energy scale �L as sketched in upper panel in Figure 2.4, because its

�-function increases with increasing coupling constant. QED is not the only theory with a Landau

pole problem. The scalar quartic coupling � in the Higgs sector has the same problem. On the other

hand, the QCD theory is a non-trivial theory, because it is an asymptotic theory which coupling is going

asymptotically to zero at high energies as sketched in the middle panel in Figure 2.4.

Finally, we have that the vacuum of the theory can be unstable if its coupling is going to negative

values at energies above an instability energy E0 as sketched in lower panel in Figure 2.4. It seems that

this vacuum instability problem appears in the SM, where the Higgs quartic self-coupling � becomes

negative at energies above ESM0 � 108 GeV to one-loop order in perturbation theory. This so-called

vacuum stability problem is investigated at the end of the section.

2.5.1 Triviality of QED

Let us start to investigate the triviality of QED. We will investigate where the Landau pole of the QED

coupling g is, i.e. at which energy scale the QED coupling blows up to in�nity. The �-function of the

QED coupling has been evaluated to fourth order of the coupling, �4, in Eq. (1.10) in Ref. [29], which is

�QED(�) � �@�

@�=�2

3�+

�3

4�2� 121

288

�4

�3; (2.82)

where �(�) � g(�)2=(4�) is the renormalized �ne structure constant which is �xed at �. We can solve

Eq. (2.81 to lowest order in � of the QED �-function. From this we get that

�@�

@�=�2

3�)

� �(�)

�(�0)

d�

�2=

1

3�

� �

�0

1

�d�) � 1

�(�)+

1

�(�0)=

1

3�ln

��

�0

�) (2.83)

1

�(�)=

1

�(�0)� 1

3�ln

��

�0

�; (2.84)

and therefore the QED running coupling is

�(�) =�(�0)

1� 13��(�0) ln

��0

� : (2.85)

If we look at the coupling in Eq. (2.85) then we can identify a pole at the momentum scale

�L = �0 exp

�3�

�(�0)

�' 91 GeV � exp

�3�

1=128

�' 10522 GeV; (2.86)

Page 25 of 193


where we have used that �(mZ = 91:19 GeV) � 1=128 (in Ref. [12]). It looks like that there is a pole at

very large energy. This pole is the famous Feldman-Landau (F-L) ghost. We can not conclude that there

is a Landau pole, because we can not use perturbation theory anymore, when the �ne structure constant

is � > 1.

The physical mechanism that works here is the phenomenon of charge screening. There will be created

virtual electron-positron pair around the bare charge. The bare charge will be surrounded of a cloud of

virtual charges which will reduces the value of the bare charge seen at large distances. The bare charge

will be more and more visible for higher and higher momentum applied. A disaster will therefore occurs

when the coupling becomes in�nite at a �nite momentum scale. It looks like that this will happen if

�(�0) is nonzero, then the pole leads to di�erent inconsistencies in the QED theory. Someone can come

to the conclusion that QED is trivial, because the theory is inconsistent except �(�0) vanishes.

This conclusion is not warranted alone, because we have excluded the higher order of the �-function in

Eq. (2.82). The �-function is also calculated perturbatively as mentioned, and therefore it is likely that

the Landau pole is an artifact of the perturbation theory. In the SM this Landau pole is at �L ' 1034 GeV

(according to Ref. [30]). In fact the F-L ghost pole does not appear before well beyond the Planck scale

(Ep = 1:22 � 1019 GeV), and therefore it seems that there is no problem.

2.5.2 Triviality of Higgs Sector

Now, we will investigate the triviality in the Higgs sector with the Higgs self-coupling � which is a scalar

�4 �eld theory. The �-function of this �eld theory is derived in Appendix E. The diagrams that contribute

to this �-function of the Higgs self-coupling � in only the Higgs sector to �rst loop-order are shown in

Figure 2.5. Terms of the Lagrangian in Eq. (2.16) that contributing to the diagrams are

�(�y�)2 =�

4h4 +

�

2(�21 + �22 + �23)h

2 + : : : ; (2.87)

where the would-be Goldstone bosons �1;2;3 also contribute in the loop diagrams.

The �rst two diagrams are the �rst loop corrections to the Higgs self-vertex, while the last diagram

is the �rst loop correction to the Higgs wave function. The �-function is equal to the second term in Eq.

(7.95) in Appendix E, which is

�� @�

@�=

24

(4�)2�2 � �0�

2: (2.88)

��

h

h

h

h h

h h

h

+ h h

h

h

� �+

Figure 2.5: The diagrams that contribute to the �-function of Higgs coupling � to �rst loop-order inHiggs sector without Yukawa couplings.

Page 26 of 193


We can derive the running coupling �(�) by solving this equation. We obtain that the running

coupling is always positive to leading order

� �(�)

�(�0)

d�

�0�2=

� �

�0

d�

�) � 1

�0

�1

�(�)� 1

�(�0)

�= ln�� ln�0 )

�(�) =�(�0)

1� �0�(�0) log��0

� ; (2.89)

which has a Landau pole. To determine this Landau pole we need to know a value of the self-coupling �

at some energy scale �0. We have from Eq. (62) in Ref. [34] the MS top-Yukawa coupling renormalized

at the top pole mass mt, which we will use in next subsection, is given by

�t(mt) =0:93587 + 0:00557� mt

GeV� 173:15

�� 0:00003

� mh

GeV� 125

�� 0:000041

��s(mZ)� 0:1184

0:0007

�� 0:00200th;

(2.90)

where there is a theoretical error, �0:00200th, which comes from non-perturbative e�ects. The Higgs

self-coupling in MS scheme which is renormalized at the pole top mass is also determined in Ref. [34]

(Eq. (63)), which is

�(mt) =0:12577 + 0:00205� mh

GeV� 125

�� 0:00004

� mt

GeV� 173:15

�� 0:00140th: (2.91)

(� = 172:44 GeV; g3 = 1:172)

Figure 2.6: Left panel: The Landau pole in the running of the Higgs self-coupling �, which diverges at�L = 1:8 � 1025 GeV. Right panel: The calculation of the value of the strong coupling at the top massg3(mt) = 1:1715 from the coupling renormalized at the Z boson mass �s(mZ) = g23=4� = 0:1184 in Table2.3.

mh [GeV] mt [GeV] mZ [GeV] �s(mZ)

125:09� 0:24 Ref. [32] 172:44� 0:60 Ref. [75] 91:1876� 0:0021 Ref. [76] 0:1184� 0:0007 Ref. [33]

Table 2.3: Physical Constants with �1� uncertainty.

Page 27 of 193


We can use this expression to determine the self-coupling to be �(mt) = 0:1260 by inserting the values

in Table 2.3 into Eq. (2.91). Therefore, the Higgs self-coupling in Eq. (2.89) will hit a Landau pole at

the energy scale

1� �0�(mt) log

��L�0

�= 0)

�L = mt exp

�1

�0�(mt)

�= 172:44 GeV � exp

�16�2

24 � 0:1260�= 8:23 � 1024 GeV: (2.92)

This Landau pole is shown in left panel in Figure 2.6. We can not know with certainty, that there is

a Landau pole, because the perturbation theory breaks down here. If the numerical calculations seems

to con�rm that the Higgs quartic coupling diverges when the Yukawa couplings vanish, then we can

conclude that the coupling � must be zero for the theory to be consistent. Thus, the Higgs sector is a

trivial theory, when the Higgs boson interacts only with itself.

2.5.3 Vacuum Stability in the SM

This problem can be alleviated by adding the Yukawa terms to the Higgs doublet term as in the SM,

such that we have from Eq. (2.16) and Eq. (2.73) the Lagrangian terms

�(�y�)2 � �t�ijqLj�it�R � �t�ijq�Li��j tR: (2.93)

We have only included the top-Yukawa coupling, because the Yukawa coupling is proportional to the

fermion mass. Thus, the top-Yukawa coupling contributes much more than the remaining Yukawa cou-

plings. The �-function for the Higgs self-coupling is derived in Appendix E, where both Higgs self- and

top-Yukawa interaction are included. The diagrams that contribute to the �-function of the self-coupling

� is shown in Figure 2.7.

�

h

h

h

h

+ � �

h

h h

h

h

h

h

h

h

h

h

h

h h

h

h

��t �t

�

+ +

+

�t

�t

�t

�t

�

Figure 2.7: The diagrams that contribute to the �-function of the Higgs self-coupling � in SM to �rstloop-order.

The �rst two loop diagrams contribute to the �rst loop order corrections to the Higgs quartic self-vertex,

while the last two diagrams contribute to the corrections to the Higgs wave function. The �-function is

Page 28 of 193


given in Eq. (7.95) in Appendix E, which is

�� @�

@�=

1

(4�)2(24�2 � 6�4t + 12��2t ); (2.94)

where the �rst term comes from the loop diagram 1 and 4, the second term comes from loop diagram 2,

while the last term comes from loop diagram 3 in Figure 2.7.

If we couple the Higgs boson to the gauge bosons, then we obtain to �rst loop-order (Eq. 3.5 in

Ref. [28])

�� @�

@�=

1

(4�)2

�24�2 � 6�4t + 12��2t � 3�g21 � 9�g22 +

3

8

�2g42 + (g21 + g22)

2��; (2.95)

where the �-function of the top-Yukawa coupling to �rst loop-order (Eq. 3.3 in Ref. [28]) is

��t � �@�t@�

=1

(4�)2

�9

2�3t �

�17

12g21 +

9

4g22 + 8g23

��t

�; (2.96)

and the gauge couplings g1, g2 and g3 are associated to the U(1)Y, SU(2)W and SU(3)C gauge symmetry,

respectively, which have following �-functions to �rst loop-order (Eq. 3.2 in Ref. [28])

�g1 � �@g1@�

=41

96�2g31 ; �g2 � �

@g2@�

= � 19

96�2g32 ; �g3 � �

@g3@�

= � 7

16�2g33 : (2.97)

Figure 2.8: The running of the Higgs self-coupling � for di�erent top masses calculated by the Matlab.The yellow line is the RG evolution of � for the top mass in Table 2.3. For the inner two lines aroundthe yellow line the top mass is varied by �1�, and the outer two lines the top mass is varied by �5�.

Page 29 of 193


According to the second term in Eq. (2.95), if the quartic coupling � is su�ciently small, then the top-

Yukawa would dominate the �-function. Therefore, it can maybe drive the quartic coupling to negative

values such that the theory becomes unstable. If the quartic coupling is very large, then the triviality

problem above might be relevant again. To studying the vacuum stability we use Matlab to solve the

coupled di�erential equations in Eqs. (2.95)-(2.97) for the quartic coupling � using Euler's method.

In Figure 2.8 the running of the Higgs self-coupling are plotted for various top masses calculated by

Matlab. The top coupling �t(mt) and the Higgs self-coupling �(mt) renormalized at the top mass are

calculated by using Eq. (2.90) and Eq. (2.91), respectively, where the values in Table 2.3 are been used.

The gauge couplings g1(mt), g2(mt) and g3(mt) renormalized at the top mass are found by calculating

the RG evolution of them as in right panel in Figure 2.6 for the strong coupling, g3. For example, the the

strong coupling at the top mass is found to be g3(mt) = 1:1715 from the coupling at the Z boson mass

�s(mZ) = 0:1184 in Table 2.3. The yellow line in Figure 2.8 is the RG evolution of � for the average value

of the top mass in Table 2.3. For the inner two lines around the yellow line the top mass is varied by �1�,and the outer two lines the top mass is varied by �5�. This plot shows that the vacuum of the SM to �rst

loop-order is unstable at energies above around 2 � 108 GeV. According to Ref. [65], the instability scale

is computed to be around 5 � 1010 GeV for two-loop QCD and Yukawa corrections with the central values

in Table 2.3. Therefore, the instability scale is pushed up, when we include the next-to-next-to-leading

order (NNLO) loop corrections.

Figure 2.9: The running of the Higgs self-coupling � for various values of the strong �ne constant andthe Higgs mass calculated by Matlab. Left panel: For the inner two lines and the outer two lines thestrong �ne constant is varied by �1� and �5� around the yellow line with the average value of �s inTable 2.3, respectively. Right panel: The Higgs mass is varied by �1� and �5� around the averagevalue in Table 2.3, respectively.

In Figure 2.9 the running of the self-coupling for various values of �s and the Higgs mass are plotted.

In left panel the inner two lines and the outer two lines the strong �ne constant is varied by �1� and

�5� around the yellow line with the average value of �s in Table 2.3, respectively. While in the right

panel the Higgs mass is varied by �1� and �5� around the average value in Table 2.3. Thus, the RG

Page 30 of 193


evolution of � is larger sensitive to the variation of the top mass than the Higgs mass and even smaller

sensitive to the strong �ne constant.

We have shown perturbatively that although in isolation the SM Higgs sector is trivial. This is

modi�ed when the top-Yukawa coupling is included. Instead, the SM is possibly vacuum unstable,

because the Higgs self-coupling, �, becomes negative at energies above its instability energy, E0 � 108

GeV as computed in the one-loop approximation.

In the upcoming sections, we will derive the �rst loop-order corrections to the mass of the Higgs boson,

where the quadratic divergent corrections give rise to a large �ne-tuning problem of the Higgs mass. This

becomes our motivation to construct an underlying model, which tries to explain the �ne-tuning of the

Higgs mass by a dynamical mechanism like in QCD.

2.6 Higgs Mass Corrections

In the quantum vacuum, there is constantly produced particle-antiparticle pairs out of the vacuum,

violating the energy conservation by taking the energy E from the vacuum for a short time t, which is

possible according to Heisenberg's uncertainty principle that says that Et < ~. These particles are called

virtual particles, and they are o�-shell (E2 � p2 6= m2).

When the Higgs boson propagates in the quantum vaccum, then it will interact with these virtual

particles. In the footnote, there is being made a simple analogy to thermodynamics as in Ref. [9], which

can help us understand the additional quantum contribution �m2h to the mass of the Higgs boson.3

The strength that the Higgs boson will interact with any SM particles is proportional to the mass of

the corresponding particle. These interactions result in corrections to the Higgs mass. The one-loop

diagrams that contribute to the Higgs mass are shown in Figure 2.10. We have the squared mass of the

Higgs boson, m2h, receives an additional contribution in the form:

m2h = m2

h0 + �m2h =m2

h0 + k�2 + � � � ; (2.98)

where mh is the physical Higgs mass, mh0 is the bare Higgs mass, and � is a cuto�. We have only

included quadratic loop-contributions, because they contribute mostly compared to the logarithmically.

We calculate later the coe�cient k to be 2:6 �10�2 from the one-loop diagrams. We have that the physical

mass of the Higgs boson is very small compared to the Planck scale (the largest cut-o� we can presently

imagine to the SM), and therefore the bare Higgs mass is needed to be �ne-tuned extremely much.

Thus, the Higgs particle is special, because there are no symmetries that protect against the quantum

�uctuations.

3We replace the quantum �uctuation of the vacuum with the thermal �uctuations of a thermodynamic system with atemperature, T . The particles, P , in the thermodynamic system play the role of the virtual particles in the vacuum, and thetemperature, T , corresponds to the cuto�, �. If we insert another particle, H, without momentum into the thermal system,then we expect that the collisions of the particles, P , will soon bring the particle, H, in thermal equilibrium. Therefore,the energy of the particle, H, will quickly become of order T . This is an analogy to what happens in the quantum system,here will the Higgs mass (analogous to H) be pushed towards � because of quantum �uctuation e�ects from the virtualparticles.

Page 31 of 193


f

p� qp

p

p� q

q q

f

h h

Z;W�

p

q q

�Z ; '�

h h

q qh h

p

h; '�; 'Z

p

W�; Z

q qh h

q qh h

W+

W�

p� qFigure 2.10: The one-loop diagrams that contribute to the correction of the Higgs mass.

In following we will calculate the amplitudes of the loop-diagrams in Figure 2.10. Let us de�ne �i�(q)as the sum of all one-particle-irreducible insertions into the propagator, i.e. that we have

1PI

�i�(q) =

Then we have that the full two-point function for the Higgs propagator is given by the geometric series

= + +

+ : : :

=�d4xhjTh(x)h(0)jieip�x

1PI 1PI

1PI

We can rewrite each Higgs propagator as i=(q2 �m2h0) and express the above series as

�d4xhjTh(x)h(0)jieiq�x =

i

q2 �m2h0

+i

q2 �m2h0

�(q)i

q2 �m2h0

+i

q2 �m2h0

��(q)

q2 �m2h0

�2

+ � � � =

i

q2 �m2h0

1Xn=0

��(q)

q2 �m2h0

�n=

i

q2 �m2h0 � �(q)

:

(2.99)

Therefore, the Higgs mass is corrected by

�m2h = m2

h �m2h0 = �(q): (2.100)

To calculating this mass correction to �rst loop-order, we need to calculate the sum of all one-particle-

irreducible diagrams for the Higgs propagator, which are shown in Figure 2.10. To this work, we will

Page 32 of 193


�rstly calculate two useful integrals (calculated in Eq. (7.197) and Eq. (7.198) in Appendix C-2). The

�rst integral is

�d4p

(2�)41

p2 �m2f + i�

=i

16�2

24�2

s1 +

m2f

�2�m2

f ln

0@� + �

q1 +

m2f

�2

mf

1A35

� i

16�2

��2 �m2

f ln

�2�

mf

��;

(2.101)

where we have used Cauchy's residue theorem to solve the integral, we have made a hard cuto� at �,

and in the last step we assume that �� mf . The second useful integral is solved as follows

�d4p

(2�)41

(p2 �m2f + i�)((p� q)2 �m2

f + i�)� � i

8�2

�1 +

� 1

0

dx ln

��

�2

��; (2.102)

where we have used following de�nitions

l � p� xq ) l2 = p2 + x2q2 � 2xqp;

� � �x(1� x)q2 + xm2 + (1� x)m2:

To solving this integral we have also carried out a Wick rotation, where we make the substitutions l0E = il0

and ~lE = ~l.

Now, we are ready to calculate the one-loop diagrams in Figure 2.10. We start with the Higgs

propagator with a fermion loop which is calculated in Eq. (7.199) in Appendix C-2. The correction gives

�i�fermion�loop =� e2

s2W

m2f

4m2W

�d4p

(2�)4Tr((p=+mf )((p=� q=) +mf )

(p2 �m2f + i�)((p� q)2 �m2

f + i�)(2.103)

�� 4e2

s2W

m2f

4m2W

i

16�2

��2 �m2

f ln

�2�

mf

��+

e2

s2W

m2f

4m2W

i

4�2(4m2

f �m2h)"

1 +

� 1

0

dx ln

��x(1� x)q2 +m2f

�2

�#;

where both the �rst and the second integral in Eq. (2.101) and Eq. (2.102) are been used. The next

diagram is the Higgs propagator with a Higgs loop, which gives the contribution

�i�h�loop =� ie2 12

3

4s2W

m2h

m2W

�d4p

(2�)4i

p2 �m2h + i�

� 3

2

e2

s2W

m2h

4m2W

i

16�2

��2 �m2

h ln

�2�

mh

��; (2.104)

where the �rst integral in Eq. (2.101) is used. There are also the same kind of diagram with both '�

loops and a 'Z loop, which give

�i�'��loop =� 21

2

i

4

e2

s2W

m2h

m2W

�d4p

(2�)4i

p2 �m2W + i�

� e2

s2W

m2h

4m2W

i

16�2

��2 �m2

W ln

�2�

mW

��; (2.105)

and

�i�'Z�loop =�1

2

i

4

e2

s2W

m2h

m2W

�d4p

(2�)4i

p2 �m2Z + i�

� 1

2

e2

s2W

m2h

4m2W

i

16�2

��2 �m2

Z ln

�2�

mZ

��: (2.106)

The next two diagrams are them with a W� and Z loop, which give

Page 33 of 193


�i�Z�loop =1

2

i

2

e2

c2ws2w

g��

�d4p

(2�)4�ig��p2 �m2

Z

� 4e2

s2W

m2Z

4m2W

i

16�2

��2 �m2

W ln

�2�

mZ

��; (2.107)

and

�i�W�loop =i21

2

i

2

e2

s2wg��

�d4p

(2�)4�ig��

p2 �m2W

� 8e2

s2W

m2W

4m2W

i

16�2

��2 �m2

W ln

�2�

mW

��: (2.108)

The diagrams with a Z='Z and a W�='� (derived in Eq. (7.200) and Eq. (7.201) in Appendix C-2)

contribute with

�i�Z='Z�loop =e2

s2W

1

4c2w

�d4p

(2�)4(�p+ q � q)�(p� q + q)�

�ig��p2 �m2

Z

i

(p� q)2 �m2Z

(2.109)

�� e2

s2W

m2Z

4m2W

i

16�2�2 + : : : ;

and

�i�W�='��loop =� 2e2

s2W

m2W

4m2W

�d4p

(2�)4(�p+ q � q)�(p� q + q)�

�ig��p2 �m2

W

i

(p� q)2 �m2W

(2.110)

�� 2e2

s2W

m2W

4m2W

i

16�2�2 + : : : :

The correction terms which are quadratic in � are interesting to consider, because they lead to a

naturalness problem. We consider the diagram where the fermion in the fermion loop is a top quark,

because it contributes much more than the other fermions because of its large mass. The quadratic

contributions are

�top =4e2

s2W

m2t

4m2W

1

16�2�2 + : : : ;

�Higgs =�h�loop +�'��loop +�'Z�loop = ��3

2+ 1 +

1

2

�e2

s2W

m2h

4m2W

1

16�2�2 + : : :

=� e2

s2W

3m2h

4m2W

1

16�2�2 + : : : ;

�Z =�Z�loop +�Z='Z�loop = �[4� 1]e2

s2W

m2Z

4m2W

1

16�2�2 + � � � = � e2

s2W

3m2Z

4m2W

1

16�2�2 + : : : ;

�W =�W�loop +�W�='��loop = �[8� 2]e2

s2W

m2W

4m2W

1

16�2�2 + : : :

=� 2e2

s2W

3m2W

4m2W

1

16�2�2 + : : : :

(2.111)

By inserting the corrections in Eq. (2.111) into Eq. (2.100), the squared mass of the Higgs boson,

m2h, receives an additional contribution

m2h = m2

h0 + �m2h =m2

h0 +e2

s2W

3

4m2W

1

16�2

�4m2

t �m2h �m2

Z � 2m2W

��2 + � � �

=m2h0 +

3GF

8p2�2

�4m2

t �m2h �m2

Z � 2m2W

��2 + � � �

�m2h0 + k�2 + � � � ;

(2.112)

Page 34 of 193


where the Fermi constant is

GF �p2e2

8m2W s

2W

: (2.113)

The result of the Higgs interactions with the virtual particles is that the squared mass of the Higgs

boson m2h receives an additional quantum contribution �m2

h = k�2, where � is the maximum energy

accessible to virtual particles and the constant k is the proportionality constant in Eq. (2.112) which is

k =3GF

4p2�2

(4m2t � 2m2

W �m2Z �m2

h) =3 � 1:1164 � 10�5 GeV�2

8p2�2h

4(172:44 GeV)2 � 2(80:39 GeV)2 � (91:19 GeV)2 � (125:09 GeV)2i

=2:57 � 10�2:

(2.114)

2.7 The EW Hierarchy Problem

If we assume that the cut-o� in Eq. (2.112) is the Planck mass � = MP = 1:22 � 1019 GeV where we at

least expect new physics, then the so-called EW hierarchy problem in the SM arises. The EW hierarchy

problem is that there is no scienti�c explanation on why the weak force is very much stronger than gravity

(the large gap between their energy scales vEW=MPlack � 10�17). We have namely that the physical mass

of the Higgs boson is very small compared to the Planck scale, and therefore the bare Higgs mass,

m2h =m2

h0 + kM2P = m2

h0 + 2:57 � 10�2(1:22 � 1019 GeV)2 = (125 GeV)2 )m2h0 =(125 GeV)2 � (2 � 1018 GeV)2; (2.115)

is needed to be �ne-tuned extremely much. In the following subsection, we will de�ne a possible quantity,

which is a measure of the �ne-tuning of an observable compared to the parameters in a model.

2.7.1 Fine-Tuning of Models

In the following, we will brie�y give a possible de�nition of �ne-tuning in models. Fine-tuning refers to

cases when the parameters of a model must be adjusted very precisely in order to agree with observations.

By de�ning a quantity for the �ne-tuning then we can compare the �ne-tuning between the various models.

This quantity which measures the amount of �ne-tuning in any particular parameter, �i, to produce the

observable, O, is historically been introduced by Barbieri and Giudice as the relative ratio between the

observable and the parameters normalized to them (cf. Eq. (36) in Ref. [45]), i.e.

�OBG;i �

��iO @O@�i

�� < �max; (2.116)

This quantity gives a measure of �ne-tuning for each parameter. One possible way to �nd the total

�ne-tuning in O could be by simply taking the maximum of all the �OBG,i. We can decide to have the

maximum value �max = 100. This means one percent change of the parameter, �i, gives rise to maximal

one hundred percents change of the observable O to have that the observable is not �ne-tuned.

Page 35 of 193


2.7.2 Fine-Tuning of the Higgs Mass

We can calculate the de�ned quantity for the �ne-tuning in Eq. (2.112) for the mass of Higgs boson, m2h,

compared to its tree-level mass, m2h0, where mh0 = mh � k�2. For a cuto� at Planck scale, � = MP =

1:22 � 1019 GeV, this quantity is

�mh

BG;mh0��m2

h0

m2h

@m2h

@m2h0

�� =��m2

h � k�2

m2h

�� =��1� k�2

m2h

�� =��1� 2:57 � 10�2 � (1:22 � 1019 GeV)2

(125:09 GeV)2

��=2:4 � 1032:

(2.117)

Thus, the �ne-tuning quantity is extremely large. Therefore, the Higgs mass, m2h, is extremely �ne-tuned

compared to its tree-level mass, m2h0.

In the following we follow Ref. [9], where it is tried to get an intuition of how much the Higgs mass

is �ne-tuned by making a simple analogy. It requires a steady hand to balance a pencil on its tip on a

table. If r is the radius of the tip surface and R is the length of the pencil, then the needed accuracy is

of the order of r2=R2. We can compared this accuracy to the �ne-tuning quantity above. By using that

the radius of the tip is about rtip � 1 mm, which gives that the length of the pencil is approximatively

�mh

BG;mh0� R2

r2tip)

R �q�mh

BG;mh0mm =

p2:4 � 1032 mm = 1:5 � 1013 m:

(2.118)

The radius of the solar system is about RSolar System = 5 � 1012 m. In Ref. [9] this �ne-tuning is compared

to that we need to balance a pencil minimum as long as the solar system on a tip of one millimeter

wide to reproduce the necessary accuracy GF =GN .4 This makes that the SM seems unnatural, among

others because this enormous �ne-tuning of the Higgs mass. The origin of the Higgs mass in the SM is

complete unclear. This indicates a need for a more general mechanism or some symmetries that provide

a rationalization for the Higgs boson. This leads us to believe that there could be a theory beyond the

SM.

It is a puzzle why the Higgs boson should be light, when the interactions between it and SM particles

would tend to make it very heavy. It can e.g be cured if the Higgs is a composite particle of some new

dynamics such that the cuto� � is low with respect to the weak scale and there is consequently only a

small amount of �ne tuning. This composite dynamics can be quite similar to QCD, where color and

quarks will be con�ned which will be observed as hadrons (mesons and baryons). This will happen below

a typical scale, the cuto� �, like QCD energy scale �QCD. Another way to cure the �ne-tuning problem

is a supersymmetric model (e.g. in Ref. [?]). Supersymmetry would link the fermions and the bosons by

a transformation that take a fermion or a boson into a boson or a fermion. Therefore, the supersymmetry

predicts a superpartner particle for each particle in the SM. Physical the loop contributions to the Higgs

mass sum to zero for energies above the supersymmetric breaking scale. The superpartner particles

predicted would cancel out the contributions to the Higgs mass from their SM partners, because their

4We have that the Higgs mass is mh � G�1=2F and the Planck mass is MP = G

�1=2N , where GF and GN is Fermi

constant and the gravitational constant, respectively.

Page 36 of 193


contributions have same size and opposite sign (Ref. [66]). Thus, the proportionality constant k in

Eq. (2.112) would become zero above the the supersymmetric breaking scale, which would remove the

�ne-tuning in Eq. (2.117).

In this thesis, we will focus on models of the Higgs mechanism, where the Higgs boson is dynami-

cally produced as either a composite or a partially composite particle consisting of new fermions, called

techniquarks. The possible dynamics that forms the Higgs boson can be similar to QCD. Thus, in the

next section we will examine the mechanism of the chiral symmetry breaking in QCD, where the quarks

con�ne to the hadrons.

2.8 Chiral Symmetry Breaking in QCD

We will examine the mechanism of chiral symmetry breaking in QCD, which dynamically generates

the masses of the hadrons. We will construct an e�ective theory for the lightest pseudoscalar-mesons

consisting of the three lightest quarks in the SM and identify their masses in the e�ective Lagrangian.

We will start by considering phenomena in the theory of the strong interaction of elementary particles,

Quantum Chromodynamics (QCD).

2.8.1 Quantum Chromodynamics (QCD)

The theory of the strong interaction of elementary particles, called QCD, is a non-abelian gauge theory

with SU(3)C as gauge group. The corresponding charges to this SU(3)C are called colors. The quarks

have besides the �avor also color and transform as the fundamental representation of color SU(3)C. The

eight colored gauge bosons, called gluons, are in the adjoint representation of SU(3)C. The de�nitions of

a fundamental and adjoint representation of a symmetry group are explained in Appendix C.

We will start writing down the Lagrangian of QCD, which is written as

LQCD = i � Ai =Dij

jA �

1

4Ga��G

��a = i � Ai =D

ij

jA �

1

2Tr[G��G

�� ]; (2.119)

where jA is the quark spinor with color index j = 1; 2; 3 and �avor index A, and the covariant derivative

can be written as

D i� j = @��

ij � igsGa�T i

a j ; (2.120)

where T ia j = � i

a j=2 are the generators of SU(3)C, i.e. �ia j are the Gell-Mann matrices with a = 1; : : : ; 8.

The gluon �eld strength tensor is

G j��i =@�G

j�i � @�G j

�i � igs[G�; G� ] ji = @�Gd� T

jdi � @�G d

� T jdi � igsGa�Gb� [Ta; Tb] ji

=@�Gc� T

jci � @�G c

� Tj

ci � igsGa�Gb�f cab T

jci = Gc��T

jci ;

(2.121)

where f cab are the structure constants. Therefore, we have

Gc�� = @�Gc� � @�Gc� � igGa�Gb�f c

ab : (2.122)

Page 37 of 193


We have written the gluon �elds as Gij = Ga�Ti

a j , wherein the two indices i; j is shifted with respect to

each other, because i is an anti-fundamental index and j is a fundamental index.

f �f

g

g

f �f

f

g

Figure 2.11: The one-loop correction diagrams to the QCD vertex.

To �rst loop-order the �-function of the QCD coupling gs (cf. (3.1.2) in Ref. [13]) is

�(gs) = � �0g3s

(4�)2+O(g5s) = �

�11� 2

3Nf� g3s(4�)2

+O(g5s); (2.123)

which is calculated from e.g. the two one-loop diagrams in Figure 2.11. We can derive the running

coupling by solving the following di�erential equation

dgsd ln�

= � �0g3s

(4�)2) �g2s =

g2s

1 + g2s�08�2 log

�Q�

� ; (2.124)

where gs = gs(�) and �gs = gs(Q). If we have that �s(�) = g2s=4� and �s(Q) = �g2s=4�, then we obtain

�s(Q) =�s(�)

1 + �02��s(�) log

�Q�

� : (2.125)

MeV

�s

�QCD

Hadrons

Con�nem

ent

Asymptotic freedom

Figure 2.12: QCD running coupling as function of the energy.

Page 38 of 193


In Figure 2.12 the running coupling �s(Q) is plotted as function of energy Q. At high energy or high

momentum transfers we have that the QCD coupling is small. We say that QCD is asymptotically free.

We can see that QCD is asymptotically free if the number of quarks is not too big (�0 > 0), when the

number of quarks is Nf < 33=2, which is met with Nf = 6 quarks in the SM. Therefore, QCD is treatable

by perturbative methods at high energy.

We can also observed that there is a speci�c energy scale, �QCD, where the coupling blows up.

Therefore, QCD becomes strongly interacting at low energy. The condition for this phenomena called

QCD con�nement is

1 = �s(�)�02�

log

��

�QCD

�: (2.126)

By inserting this condition into Eq. (2.125) we obtain this form of the QCD coupling

�s(Q) =2�

�0 log�

Q�QCD

� ; (2.127)

where �0 = 11 � 23Nf = 7 in the SM, because there are six quarks in the SM. Thus, we have that the

running constant of QCD grows for decreasing energy. By dimensional transmutation the interaction

may be characterised by the dimensionful parameter, �QCD, namely the value of the RG scale at which

the coupling constant diverges. Dimensional transmutation is a physical mechanism providing a linkage

between a dimensionless parameter (e.g. the QCD coupling gs) and a dimensionful parameter (the

energy scale �QCD). Below this QCD scale, a con�nement of quarks and gluons in hadrons happens

below around 1 GeV. Perturbation theory, which produced the running formula above, is only valid for

a coupling �s � 1. According to lattice calculations and experiments, there exists such a QCD scale,

which is �QCD = 217+25�23 MeV (in Ref. [37]), which is an infrared cuto� (Q� �QCD implies �s � 1). The

masses of the hadron resonances are in the order of this scale. The energy scale, �, in Eq. (2.112) can

maybe have a natural origin relative to the Planck scale, if it is explained by a dynamical mechanism as

the QCD scale above. On the other hand, in the case of theories such as QED, � is an ultraviolet cuto�

(Q� � implies �� 1) at which the Landau pole happens as shown in left panel in Figure 2.6. However,

it seems that the Landau pole takes place long above Planck scale, therefore there are no problems.

The hadron dynamics at low energies can be investigated by performing nonperturbative numerical

computations, e.g. by lattice QCD or by the method of e�ective �eld theories. The e�ective �eld theories,

e.g. the non-linear � model, is based on at low energies a description of the strong interaction directly

in terms of the light hadrons, e.g. the pseudoscalar mesons, is possible. Now, we will describes how to

construct such e�ective �eld theories.

2.8.2 Construction of an E�ective Lagrangian

In this subsection, we will talk about how we can construct an e�ective Lagrangian, which can describe

the composite particles consisting of quarks in QCD or the Higgs boson as composite particle consisting of

techniquarks. When studying a physical system it is often the case that there is not enough information

Page 39 of 193


about a fundamental description of some of its properties, e.g. when the perturbation theory breaks down

at low energies in QCD. In these cases we need to parameterize the corresponding e�ects by writing new

interactions with coe�cients which can be determined phenomenologically. Experimental measurements

of these parameters can hopefully provide the information needed to provide a better description of the

properties of the model.

For doing this, we need to determine the dynamically degrees of freedom involved, and the symmetries

they obeyed. Thereafter, we construct the e�ective Lagrangian for these degrees of freedom which will

respect the required symmetries. It is important to have in mind that the relevant degrees of freedom

can change with energy scale (e.g. mesons are a good description of low-energy QCD, but not at higher

energies where we need to use quarks and gluons), and the physics can respect di�erent symmetries at

di�erent energy scales. Thus, the e�ective Lagrangian is applicable only for a limited range of energy

scales. It is often that there is an energy scale �, where for energies above � the e�ective Lagrangian

does not work anymore. This method of e�ective theories is straightforward, and most importantly it

works.

To begin with, we can concentrate us about the Lagrangian of QCD (strong interactions), which (cf.

Eq. (2.119)) can be written

LQCD = i � Ai =Dij

jA �

1

4Ga��G

��a ; (2.128)

where Ga�� is the gluon �eld strength tensor and Ai are the quark �elds. We can write such a compact

form of QCD because of gauge invariance and renormalization. As mention, the quarks com�nes hadrons

(mesons and baryons) below an energy about one GeV, which is set by the QCD cuto�, �QCD � 200MeV.

At this energy scale, it is not possible yet to calculate QCD results exact, because the QCD coupling

constant has been too large that we can use perturbation theory anymore. Therefore, we want to make

an e�ective theory for these composite particles.

In this case we are interested in the description of the interactions among the lightest composite

particles (e.g. the mesons in QCD). The most convenient parameterization of these degrees of freedom

is in terms of the nonlinear unitary �eld (exponential parameterization) such that

U(x) = exp

�i�a(x)Xa

f

�; (2.129)

where �a(x) denote the Goldstone �elds (the lightest meson �elds in QCD) coming from the global

spontaneously breaking pattern G ! H, Xa are the generators of the broken global symmetries, f is a

constant (related to the pion decay constant in QCD), which has unit of energy and therefore makes the

argument of exponential unitless. We have also that UUy = 1n, because the generators are hermitian.

The quantity U transform under G global group as U ! gUgy, where g 2 G. The e�ective Lagrangianmust obey this global symmetry G of the fundamental Lagrangian, gauge symmetry invariance, Lorentz

invariance, C invariance and P invariance. The e�ective Lagrangian with a cuto� as in QCD, there

is no actual ultraviolet divergences in most e�ective Lagrangian computations. Therefore, we have not

necessarily the renormalization as guide line to constructing the e�ective Lagrangian terms.

Page 40 of 193


With these constraints the e�ective Lagrangian takes the form

Le� =Tr[@�Uy@�U ] + Tr[U@�U

yU@�Uy]+

Tr[@�U@�Uy@�U@�Uy] + In�nite Many Terms;

(2.130)

which is invariant under the global symmetry, Lorentz, C and P transformations. There can not be

derivative-free terms, because Tr[UUy] = Tr[1n] = n is a constant. We have the derivative @� is p� in

momentum space, and it follows from dimensional analysis that the coe�cient of an operator with k

derivatives behaves as 1=�k�4, whre � is a mass scale which depends on the speci�c theory. Thus, the

e�ect of an n-derivative vertex is of order pk=�k�4, and thus at an energy small compared to �, then

large-k terms have a very small e�ect. Therefore, the in�nite many terms would become less and less

important, and then we can make a perturbative expansion in derivative at su�ciently low energies. In

next subsection, we will construct such a low-energy e�ective theory for the eight pseudoscalar mesons,

�a, in QCD consisting of the three lightest quarks in the SM.

Problems happen when we have theories with heavier particles than the Goldstone bosons, other

scalar excitations. In this case, we can arrange the �elds as follows

U(x) = S(x) exp

�i�a(x)T a

f

�(2.131)

with the heavier �elds S(x), which gives us that UUy = SyS 6= 1n. This gives terms which are derivative-

free, because they are not constant in this case. Therefore, we have the extra terms with the form

Xn

Tr[UUy]n: (2.132)

This is a sum of in�nite many terms, where each term is equally important. Therefore, it breaks down

when we add the scalars together with the Goldstone bosons. For example, we can construct a model

like QCD with a composite Higgs consisting of smaller constituents, where Higgs is a scalar particle.

For example, we can substitute the M matrix into the U place. In this case, such a model will produce

vertices with many Higgs �eld, h, external lines. However, we can ignore many-h-vertices: Firstly it is

hard to produce, and secondly at a given number of h in the vertex then the energy would be above the

energy scale where h would fall apart.

Now, we will construct such an e�ective �eld theory in QCD for the lightest pseudoscalar mesons

which consist of the three lightest quarks (the up, the down and the strange quark).

2.8.3 Chiral Symmetry Breaking

The connection between the fundamental QCD Lagrangian and the low-energy e�ective theories is con-

structed by symmetries of the sector of the lightest quarks (e.g. the three lightest u,d and s quarks), which

appear when masses of these quarks vanish (the chiral limit). The mass term in the QCD Lagrangian is

LQCD,m = �Xq

mq� q(x) q(x); (2.133)

Page 41 of 193


where mq is the masses of the quarks q = u; d; s; � � � . If the Nf quark masses are equal, then the

Lagrangian is invariant under a SU(Nf )V symmetry ! exp(i�a�a=2) , where �a are the SU(Nf )

generators. This symmetry leads to the conserved vector currents

ja�(x) =� (x) �

�a

2 (x): (2.134)

If the quark masses vanish we have also that the axial SU(Nf )A transformations ! exp(�a 5�a=2)

is a symmetry. This symmetry leads to the conserved axial-vector currents

ja5;�(x) =� (x) � 5

�a

2 (x): (2.135)

We have also the singlet vector (U(1)V symmetry) and axial-vector currents (U(1)A symmetry)

j0�(x) =� (x) �1 (x); (2.136)

j05;�(x) =� (x) � 51 (x); (2.137)

where 1 is the unit vector in quark �avor space. These global symmetries are still symmetries, even

though that the quark masses are di�erent.

The charges of these currents (in Eqs. (2.134)-(2.137)) generate the global group

SU(Nf )V SU(Nf )A U(1)V U(1)A = SU(Nf )L SU(Nf )R U(1)L U(1)R; (2.138)

where it is very convenient to consider besides vector and axial currents also the chiral currents,

jL;�(x) =1

2[j�(x)� j5;�(x)]; jR;�(x) =

1

2[j�(x) + j5;�(x)]; (2.139)

j0L;�(x) =1

2[j0�(x)� j05;�(x)]; j0R;�(x) =

1

2[j0�(x) + j05;�(x)]; (2.140)

which have the symmetries SU(Nf )L, SU(Nf )R, U(1)L and U(1)R, respectively.

The global �avor symmetry of the QCD Lagrangian in Eq. (2.119) is

U(Nf )L U(Nf )R �=SU(Nf )L SU(Nf )R U(1)L U(1)R (2.141)

=SU(Nf )V SU(Nf )A U(1)V U(1)A: (2.142)

where SU(Nf )V gives conservation of the strong isospin, and U(1)V gives the conservation of the baryon

number. When we quantize the theory then the global group U(1)A is broken, because it is not anomaly-

free (i.e. the measures in the Feynman path integrals are not invariant under these transformations).

The rest of the chiral symmetry with dimension 2N2f � 1 is spontaneously broken to a subgroup of only

the vector symmetries with dimension N2f as follows

SU(Nf )V SU(Nf )A U(1)V �! SU(Nf )V U(1)V: (2.143)

This global symmetry of the Lagrangian with the quark mass terms is an approximatate symmetry,

because the quarks have masses. For the lightest three quarks, we can assume that they are approximately

Page 42 of 193


massless, because their masses are somewhat less than the QCD scale �QCD. For the up, down and strange

quark we have

mu

�QCD;md

�QCD� 1 and

ms

�QCD< 1: (2.144)

If the quark masses are di�erent then the global symmetry is only U(1) U(1), because the isospin

symmetry SU(Nf )V is broken to U(1). Because of Eq. (2.144) we have that the QCD Lagrangian has

the approximate global symmetry in Eq. (2.143) for the three lightest quarks (i.e. Nf = 3).

The Vafa-Witten theorem shows that vector global symmetries such as strong isospin (SU(Nf )V

charge) and baryon number (U(1)V charge) in vectorial gauge theories like QCD cannot be spontaneously

broken (cf. Ref. [46]). Therefore, the vectorial symmetries are unbroken after the spontaneously breaking

in Eq. (2.143). According to this theorem, the spontaneous breaking can maximally break SU(Nf )A,

which generates N2f � 1 pseudoscalar Goldstone bosons. E.g. in the case where there are three massless

quarks, then we can identify eight pseudoscalar Goldstone bosons, �+, �0, ��, K+, K0, �K0, �K� and �8.

The explicit breaking of chiral symmetry by the QCD quark mass terms will generates their experimentally

observed masses.

The amplitude of the production of a pseudoscalar meson state jp; bi from the vacuum j0i (cf. page509 in Ref. [13]) is

h0jja5�jp; bi = ip�faPS�

ab; (2.145)

where a; b = 1; : : : ; N2f � 1. This transition amplitude contains the order parameter, the pseudoscalar-

meson decay constant faPS , which is non-zero if the global symmetry is broken.

The e�ective Lagrangian can be expanded in a serie in the number of the pseudoscalar mesons �elds

(�a �elds) as in Eq. (2.130) as follows

Le� = L(2)e� + L(4)e� + � � � : (2.146)

which corresonds to an expansion in the momentum in momentum space. The �rst term is

L(2)e� =1

2@��

a@��a: (2.147)

There can not be constructed a �3 term which is Lorentz- and �avor-invariant. Therefore the �rst

interaction term is a �4 term.

In the following, we construct an e�ective Lagrangian for the eight pseudoscalar mesons consisting of

the three lightest quarks. The e�ective Lagrangian can be expressed in terms of the exponentials of the

� �elds as in Eq. (2.129). This model is called the non-linear � model. The �elds are written as

U(x) = exp

�i�(x)

fPS

�; (2.148)

where we have the �eld

Page 43 of 193


�(x) = �a(x)�a =

0BBB@

�3 +�8p3

p2�+

p2K+

p2�� 3 + �8p

3

p2K0

p2K� p

2 �K0 � 2�8p3

1CCCA : (2.149)

The broken generators, �a, are the eight Gell-Mann matrices. The Lagrangian of the non-linear � model

is constructed from U(x) in Eq. (2.148) such that it reproduces the term in Eq. (2.147). This �ts with

L� =f2PS4

Tr[(@�U)(@�Uy)]; (2.150)

which is invariant under the transformation

U 0(x) = gLU(x)gyR; (2.151)

where gL and gR are the elements of SU(Nf )L and SU(Nf )R groups, respectively. We can expand the

L� further to higher-order terms. The quartic term is

L(4)� =Xi

LiPi =L1�Tr[(@�U)(@

�U)y]�2

+ L2Tr[(@�U)(@�U)y]Tr[(@�U)(@�U)y]+

L3Tr[(@�U)(@�U)y(@�U)(@�U)y] + � � � ;

(2.152)

where the Li are free parameters. In reality the masses of the light quarks do not vanish. The chiral

symmetry is broken explicitly by the QCD mass term in Eq. (2.133). We induces a corresponding term

in the non-linear � model. A suitable ansatz can be

L�;SB =f2PS4

2B0Tr[Mq(U + Uy)]; (2.153)

where B0 is a free constant and Mq is the mass matrix of the light quarks. By expanding the exponential

of the pseudoscalar �elds U(x) (Eq. (2.148)) in the symmetry breaking term (Eq. (2.153)) to second

order and leaving out the constant term gives the mass term of the pseudoscalar mesons,

L�;SB =f2PS4

2B0Tr[Mq(U + Uy)] =f2PS4

2B0Tr

�2 +

�a(x)�a�b(x)�b

f2PS+O(�4)

�)

L�;M =�B0

�mu

��232

+ �+�� +K+K� +�286

+�3�8p

3

�(2.154)

+md

��232

+ �+�� + �K0K0 +�286� �3�8p

3

�+ms

�K+K� + �K0K0 +

2�283

��:

After diagonalization of the mixing terms of the �elds �3 and �8 in Eq. (7.202) in Appendix C-2, we

obtain the masses

M2�� =(mu +md)B0; M2

K� = (mu +ms)B0; M2K0 = (md +ms)B0;

M2�0 =

�mu +md � (mu �md)

2

2(2ms �mu �md)

�B0 +O

�(mu �md)

3�; (2.155)

M2� =

�mu +md + 4ms

3+

(mu �md)2

2(2ms �mu �md)

�B0 +O

�(mu �md)

3�:

Page 44 of 193


The expectation value of the mass term in Eq. (2.153) is

Df2PS2B0Tr[Mq(U + Uy)]

E=f2PS2B0Tr[2Mq13] = f2PSB0(mu +md +ms); (2.156)

and according to Eq. (2.133) we have that the expectation value of the mass term at fundamental level

is

D�Xq

� qmq q

E= �muh � u ui �mdh � d di �msh � s si: (2.157)

By setting these mass terms equal each other, we obtain

muh � u ui+mdh � d di+msh � s si = �f2PSB0(mu +md +ms))h0j � q qj0i = �f2PSB0 no sum over q:

(2.158)

We have the quark condensate is related to the squared of the GB masses MGB in the Gell-Mann-Oakes-

Renner (GMOR) relation (cf. Eq. (1) in Ref. [47]), which is written as follows

M2GBf

2PS = �

Xq

h � q qimq: (2.159)

By inserting Eq. (2.158) into GMOR relation above we obtain that the masses of the pseudoscalar mesons

are

M2GB =

Xq

mqB0; (2.160)

which are consistent with the lightest GB masses in Eq. (2.155).

In the next section, we will transfer the way to produce the masses of Goldstone bosons to a simple

technicolor model. It could be a QCD-like theory with typical energy scales in the order of TeV with

bound states of new kind of fermions which provide the SM Higgs boson. In these models, the SM Higgs

boson achieves its mass from a dynamical mechanism like in QCD, where the masses are only a�ected

logarithmically by quantum corrections. Thus, the EW hierarchy problem would not exist.

2.9 Technicolor Models

The idea of technicolor is that the EW hierarchy problem associated with the mass of the Higgs boson

can be evaded if the Higgs boson is not an elementary particle but a composite object. If this object are

made of constituents which have masses only a�ected logarithmically by quantum corrections, then the

EW hierarchy problem would not exist. Such models require that the interactions are strong such that

the Higgs can be a bound state, and therefore we need to apply non-perturbation theory. Thus, the Higgs

boson would appear as an elementary particle only at energies signi�cantly below its binding energy.

This construction is actually rather intuitive, because a similar construction is realized in the SM

already. In QCD, there are bound states of quarks as considered in previous section which have the same

quantum numbers as the Higgs and can induce the breaking of the EW symmetry. The QCD condensates

Page 45 of 193


with the quantum numbers of the Higgs condensate can be constructed, but the challenge is that the

QCD scale (�QCD � 200 MeV) is not su�cient to provide the observed breaking of the EW symmetry

(vEW = 246 GeV).

Therefore, the simplest extension to the SM could be a QCD-like theory with typical energy scales

in the order of TeV with bound states which provide the SM Higgs boson. These theories are called

technicolor theories.

2.9.1 Simple Technicolor

The simplest version of technicolor model is a QCD-like theory at higher energy scale. In this simple

technicolor model, we have a gauge group SU(NTC) with Nf additional fermions Q, called techniquarks.

These techniquarks are placed in the fundamental representation of the gauge group SU(NTC), which

are massless at tree-level. Thus, there are NTC technicolors. In addition, there are the N2TC � 1 gauge

bosons, called technigluons. Therefore, the total gauge group which is an extension of the standard

model is SU(NTC)TC SU(3)C SU(2)W U(1)Y. Such a theory looks very much like QCD, except it

may have possibly a di�erent number of colors and �avors. Its dynamics will be quite similar to QCD,

where technigluons and techniquarks will be con�ned, and techniquarks can only be observed bound in

technihadrons. This dynamics will be determined by a typical scale, e.g. in QCD this scale is �QCD � 200

MeV. In the technicolor model there exists also such a scale �TC. This scale must be of the same size as

the electroweak scale, otherwise the EW hierarchy problem will emerge again. We assume that this scale

is of the size of 1 TeV instead of 1 GeV like in QCD.

The techniquarks are so far approximately massless similar with QCD. Because they are massless

then there is a chiral symmetry that generates the global group SU(Nf )V SU(Nf )A U(1)V U(1)A.

As in QCD, because of the dynamics of the technigluons the chiral symmetry of the techniquarks is

spontaneously broken to the global group SU(Nf )V U(1)V. This gives us N2f � 1 goldstone bosons

similar to pions and the other pseudoscalar mesons in QCD which are pseudoscalar bound states of a

techniquark Q and an anti-techniquark �Q. The condensate will have a size of about �TC as we can

observe in QCD, which give the techniquark an e�ective dynamical mass of the order of �TC.

If the techniquarks have quantum number as the ordinary quarks, the technipions will just have the

same quantum numbers such that they can become the longitudinal degrees of freedom of the weak gauge

bosons instead of the would-be bosons in the Higgs mechanism. The Higgs is not one of the Goldstone

bosons, but it will be a scalar meson which is the analogue of the � meson of QCD. It is expected to be

more massive and also more unstable than the Goldstone mesons.

Since a technicolor theory is based upon an analogy with the dynamics of QCD, then we can rescale

QCD to determine the properties of the pure technicolor theory. The main scaling rules (Eq. (2.30) in

Ref. [15]) are

fQCDPS �pNC�QCD; hQiQjiQCD � �ijNC�

3QCD; m0 � �QCD; (2.161)

Page 46 of 193


where hQiQjiQCD is the QCD condensate, fQCDPS � 93 MeV is the pseudoscalar-meson decay constant in

QCD in Eq. (2.145), NC is the number of colors, �QCD � 200 MeV is the QCD energy scale, and m0 is

dynamically generated constituent mass. From the �rst scaling rule above we can obtain a relation for

the technipion decay constant, which is

fTCPS �rNTC

3

�TC�QCD

fQCDPS : (2.162)

The TC e�ective gauge-kinetic Lagrangian as in Eq. (2.150) takes the form

QLi =DQL +QRi =DQR !�fTCPS

�24

Tr[D�U(D�U)y]: (2.163)

whereQL;R are the left-handed and right-handed the techniquarks, respectively. Therefore, ifND doublets

carry weak charges, then there are ND terms of the form above. Thus, the weak scale becomes vEW =pNDf

TCPS . By inserting the TC decay constant in Eq. (2.162) into this expression for the weak scale, we

obtain

vEW �rNDNTC

3

�TC�QCD

fQCDPS : (2.164)

From this expression we can isolate the TC energy scale and determine the TC decay constant

�TC �r

3

NDNTC

vEW

fQCDPS

�QCD �r

3

NDNTC

0:7 TeV; fTCPS � vEW

r1

ND; (2.165)

where vEW = 246 GeV, �QCD � 250 MeV and fQCDPS = 92:3 MeV (in Ref. [38]). With these scaling rules,

we can determine the key properties of the main classes of TC theories.

As we will see, the Goldstone bosons may not be used as the Higgs boson in a technicolor model.

Therefore, we need to identify the Higgs boson as the � scalar in QCD. In the following, it gives rise to

a problem, because we can not produce a techni-� with the Higgs mass mh = 125 GeV. We have that

the techni-pion decay constant (fTC� ) is the same as the EW energy scale (vEW) for one doublet. We can

determine fTC� by scaling the pion decay constant constant in QCD up to EW scale with the scaling rule

in Eq. (2.162),

fTC� = vEW = 246 GeV = �fQCD� ; (2.166)

where we expect fTC� / pNTC and fQCD� / pNC according to the scale rule in Eq. (2.161), where NTC

and NC = 3 are the number of colors in the TC model and colors in QCD, respectively. By knowing that

fQCDPS = 92:3 MeV (cf. Ref. [38]), we have

� = 2665 �r

NC

NTC

; (2.167)

if the technicolor guage group is SU(NTC). Thus, the Higgs boson can be compared to the lightest

composite scalar state f0(500) or also called � in QCD, such that the Higgs is a techni-�. In this case

the mass of the Higgs boson scales as follows (as in Ref. [11])

mh = �m� = (1070 to 1470)

r3

NTC

GeV; (2.168)

Page 47 of 193


where the mass of the scalar meson f0(500) or � is m� = (400� 550) GeV (cf. Particle Data Group). By

rescaling this resonance in QCD to technicolor in Eq. (2.168), it is too heavy to be the observed Higgs

boson with 125 GeV mass, unless the number of technicolor is very high. This in turn is constrained by

electroweak precision measurements as follows in the next chapter.

This is one of the di�culties with the Higgs mechanism as a technicolor model. In the following

chapters, we will experience that there are other problems with the technicolor models than this one, e.g.

the generation of the SM fermion masses in the composite dynamics, �avor-changing neutral currents

and the construction of the mass hierarchy between the fermions.

2.10 Chapter Conclusion

We can conclude that the Higgs boson is responsible for the origin mass of the elementary particles in

the SM and for curing the would-be violation of unitarity in the weak sector. We can also conclude, that

the custodial symmetry is minimal broken by the Yukawa sector in the SM. According to experimental

data from LEP experiments, the parameters describing the unitarity and the breaking of the custodial

symmetry are �W = 0:91+0:10�0:12 and � = 1:0006� 0:0009, respectively. These measurements are consistent

with the SM, where both parameters are normalized to be one. Firstly, we can conclude, there are

still room to a new Higgs doublet or other physics beyond the SM which is responsible for unitarity

of the scattering amplitudes. Secondly, we have that new physics beyond the SM should be custotial

symmmetric or the symmetry must be broken minimal. Furthermore, we can conclude that the SM

model seems to be a non-trivial theory without Landau poles and may to be unstable with an instability

energy at E0 � 108 GeV to one-loop perturbative calculations.

The SM may not be a complete theory of the EW symmetry breaking according to the calculations of

the Higgs mass corrections, because the Higgs mass is very �ne-tuned. Our main motivation is to search

after a possible dynamical origin of the EW scale which would be natural. Thus, we tried simply to

construct a rescaled QCD model. In these models the Higgs boson is a composite resonance consisting of

techniquarks like the hadrons in QCD. These kind of models have a natural cuto� scale, which is explained

by an underlying dynamical mechanism like the energy scale �QCD in QCD. One of the di�culties to

establish these technicolor models is that it is heard to explain the mass of the observed 125 GeV Higgs

boson, unless the number of technicolor is very high. As follows in the next chapter, the electroweak

precision measurements at the LEP experiments constrain the possible number of technicolor.

Page 48 of 193

Chapter 3

Minimal Walking Technicolor

The Minimal Walking Technicolor (MWT) theory is proposed in Ref. [4] and e�ective Lagrangian is

developed in Ref. [5], where we would mostly follow Ref. [1] in this chapter. In this theory, we have the

extended gauge group SU(2)TCSU(3)CSU(2)WU(1)Y. The �elds of the technicolor SU(2)TC gauge

group are the technifermions UL, DL, UR and DR, and technigluons which all transform according to the

adjoint representation of SU(2)TC as described in Appendix C.

The composite section of the model su�ers from the Witten topological anomaly, because there are

an odd number of left-handed fermion doublets under the weak gauge group, since there are three

QL � (UL; DL) doublets. The Witten topological anomaly is explained in Appendix H. The model

is cured by adding a new fermionic weak doublet LL, which are singlets under technicolor gauge group.

Furthermore, the weak singlets NR and ER are introduced to cancel the gauge anomalies with the hyper-

charge assignment in Table 3 as shown in Appendix B, where the parameter y can take any real value.

We refer to the �elds as LL, NR and ER the New Leptons.

Field SU(2)TC SU(3)C SU(2)W U(1)Y

QL =

0B@ UL

DL

1CA 3 1 2 y

2

UR 3 1 1 y+12

DR 3 1 1 y�12

LL =

0B@ NL

EL

1CA 1 1 2 �3y2

NR 1 1 1 �3y+12

ER 1 1 1 �3y�12

Table 3.1: Representations of fermions in MWT under SU(2)TC, SU(3)C, SU(2)W and U(1)Y.

In the analysis of the e�ective Lagrangian with the global symmetry SU(4), we assume that this SU(4)

49

CHAPTER 3. MINIMAL WALKING TECHNICOLOR

symmetry spontaneously breaks to SO(4), because the condensate hURUL + DRDLi is only invariant

under SO(4) � SU(4). In this model the EW symmetry breaks simultaneously with the chiral symmetry,

because we can �nd SU(2)W U(1)Y � SU(4) and U(1)Q � SO(4). In this chiral symmetry breaking

there is also found a triplet of GBs which are absorbed as the longitudinal components of the weak

gauge bosons like in the Higgs model. Additionally, there are six Goldstone bosons. The lightest scalar

excitation around the vev can be identi�ed with a possible Higgs candidate as in the simple TC model

in previous chapter. Finally, we can identify a custodial symmetry in the unbroken symmetry group

SU(2)C � SO(4).

After we have constructed the e�ective Lagrangian and derived the spectrum of the masses of the

composite scalars and vectors in the theory, we discuss how the fermions obtain their masses from an

extended technicolor (ETC) theory with a higher symmetry group SU(NETC). Subsequently, it is shown

that a walking theory can alleviate the potential problems which such an ETC theory creates. In such

a theory the coupling of the gauge theory is nearly constant from the scale �TC to �ETC, which is

determined by the number of colors and fermion �avors if the theory is a QED-like, QCD-like, Banks-

Zaks or Walking theory. The walking dynamics was �rst introduced in Refs. [21�25]. Therefore, the theory

is called Minimal Walking Technicolor, minimal because we have the minimal number of technifermions

gauged under the EW group (only two technifermions), walking because the coupling is constant in a

wide range of energy, and technicolor because the vacuum of the theory is aligned in the technicolor

limit where the EW symmetry breaking happens at the same scale as the chiral symmetry breaking. In

principle, it is possible to have more fermions which have no EW interactions, and such that it does not

contribute to the EW precision parameter, the S parameter, cf. Ref. [26].

At the end of the chapter, we provide the link between the theory at the underlying and e�ective

Lagrangian level via the Weinberg sum rules (WSRs) in the case of running or walking dynamics by

following Ref. [26, 40, 41]. Running dynamics means here that the coupling has a dependence of energy

which is similar the one in QCD or MWT, i.e. asymptotically free gauge theories. In the walking

dynamics the coupling is nearly constant for a wide range of energy thereafter the running behaviour

resumes at high energies. In same section, we derive a formula for the EW parameter called S, which can

be calculated from the underlying Lagrangian and thereafter be compared with the experimental limits

of the S parameter.

3.1 The Underlying Lagrangian for Minimal Walking Technicolor

We consider a new dynamical sector which is an SU(2)TC technicolor gauge theory with two tech-

nifermions. The two technifermions, which are in the adjoint representation of the SU(2)TC technicolor

gauge group, can be written as

U =

0@ UL;�;a

U _�;aR

1A and D =

0@ DL;�;a

D _�;aR

1A ; (3.1)

Page 50 of 193


where �; _� = 1; 2 are Lorentz indices, and a is the adjoint gauge index of the gauge group of the theory.

We have the Weyl spinors UL;�;a and DL;�;a which are the left-handed techniup and technidown, and

U _�;aR andD _�;a

R which are the corresponding right-handed particles with technicolor index a. According to

Figure 3.5 explained later, we must have that �0 < 0 in Eq. (3.109) to get a theory which is asymptotically

free. We have that

�0 =4

3TR � 11

3CA < 0) Nf < 11=4 = 2:75; (3.2)

where CA = 1 and TR = Nf=2 when the technifermions and technigluons are in the adjoint representation

of SU(2)TC. Thus, the theory is asymptotically free if the number of �avors is less than 2:75, which is

the case in this theory.

The two left-handed adjoint techniquarks and the two right-handed techniquarks can be written in

a doublet and two singlets of the EW group as Dirac spinors instead of Weyl spinors in Eq. (3.1),

respectively,

QaL =

0@ UaL

DaL

1A ; UaR; Da

R; a = 1; 2; 3; (3.3)

where a is the adjoint color index of the gauge group SU(2)TC. These left-handed techniquarks are

arranged in three weakly charged doublets. So far, the model su�ers from the Witten anomaly according

to Appendix H. We have that an SU(2)TC gauge theory is mathematically inconsistent if there are an

odd number of left-handed doublets and no other representations in this theory. However, this can be

solved by adding a new weakly charged leptonic doublet and their right-handed singlets, which can be

written as

LL =

0@ NL

EL

1A ; NR; ER; (3.4)

which are technicolor singlets. Therefore, now we have a total of four weakly charged doublets, which

removes the Witten anomaly.

It is convenient to use the Weyl basis for the fermions in Eq. (3.1) and arrange them in a vector

that transforms according to the fundamental representation of SU(4). First, we will de�ne following

left-handed spinors,

~UL;�;a �Vab�" _� _�U

_�;bR

��= Vab"��

�U

_�;bR

��= Vab"��

�U�R��;b

; (3.5)

~DL;�;a �Vab�" _� _�D

_�;bR

��= Vab"��

�D

_�;bR

��= Vab"��

�D�R��;b

; (3.6)

such that we can construct a vector in �avor space which transforms uniformly under Lorentz transfor-

mations and gauge transformations as a left-handed �eld in the adjoint of SU(2)TC.

We can construct the vector

Page 51 of 193


QAL;�;a =

0BBBBBB@

UL;�;a

DL;�;a

Vab"��(U�R)�;b

Vab"��(D�R)�;b

1CCCCCCA�

0BBBBBB@

UL;�;a

DL;�;a

~UL;�;a

~DL;�;a

1CCCCCCA; (3.7)

where A = 1; : : : ; 4 is an SU(4) index. The possible kinetic terms of the left- and right-handed Weyl

spinors are

i�UyL�_�� _��@�UL� and

i�UyR�� _�@�U

_�R;

(3.8)

where all Lorentz indices are contracted, and we can also construct similar kinetic terms for the tech-

nidown. These are invariant under transformations of the SU(2)L or SU(2)R group for the left- and

right-handed term, respectively. We can make the theory to a gauge theory by making the substi-

tution of the covariant derivative instead of the partial derivative. I.e. we make the substitution

@� ! Dab� = @��

ab + igTCAi�T

i;ab, where Ai� are the gauge �elds, T iab are the generators of the gauge

group, and gTC is the technicolor coupling constant.

Therefore, the kinetic terms for the left- and right-handed techniup and technidown can be written

as follows

LK =i�UyL�_�;a

�� _��Dab� UL;�;b + i

�DyL�_�;a

�� _��Dab� DL;�;b+ (3.9)

i�UyR��;a

�� _�Dab� U

_�;bR + i

�DyR��;a

�� _�Dab� D

_�;bR ;

which is invariant under transformations of the SU(2)L SU(2)R symmetry group, and where Dab� =

@��ab + igTCA

i�T

iab is the covariant derivative. This kinetic Lagrangian with the SU(2)L SU(2)R sym-

metry can be written instead as a kinetic Lagrangian with SU(4) symmetry which consists of the QAL;�;a

�eld (de�ned in Eq. (3.7)). As shown in eqs. (7.204)-(7.213) in Appendix C-3, such a kinetic Lagrangian

can be written in terms of the Q vector as follows

LK = i�QyAL

�_�;a

�� _��Dab� Q

AL;�;b =iQ

yAL; _�;a��

� _��Dab� Q

AL;�;b

=iQyAL; _�;a�� _��@��

ab + igTCAi�T

i;ab�QAL;�;b:

(3.10)

The second term in the covariant derivative in Eq. (3.10) can be rewritten by using that if we have an

unitary transformation in Eq. (7.211),

V �1TiV = �(Ti)�; (3.11)

then for V = I such that Ti = �(Ti)� for every i we have that the representation R is real. If V 6= I, we

have that the representation R is pseudoreal. If such unitary matrix does not exist, the representation R

is complex. If Ti is in the complex representation in Eq. (7.210), then we have not the SU(4) symmetry,

because we can not perform the last step in Eq. (7.210). We have instead only the SU(2)L SU(2)R

Page 52 of 193


symmetry as in the kinetic Lagrangian in Eq. (3.9). In our case, the gauge group is a SU(2)TC gauge

group, which is in the pseudoreal representation, therefore there is a SU(4) symmetry.

In the following we will derive an expression of the mass term in terms of the SU(4) vector Q. Assuming

the SU(4) symmetry spontaneously breaks to SO(4) or Sp(4). The mass terms have the form

�URUL + �DRDL = �1

2QAL;�;aQ

B;�;aL E�AB = �1

2QTE�Q; (3.12)

which is derived in Eq. (7.214) in Appendix C-3, and where the vacuum matrix is

E� =

0@ 0 1

�1 0

1A =

0BBBBBB@

0 0 1 0

0 0 0 1

�1 0 0 0

0 �1 0 0

1CCCCCCA: (3.13)

We have E+ in the mass term if the matrix V ab is symmetric (V ab = V ba), and E� if it is antisymmetric

(V ab = �V ba) as shown in Eqs. (7.214)-(7.221) in Appendix C-3.

We can show that a spontaneous breaking of the global SU(4) symmetry to SO(4) or Sp(4) (for E+

or E� respectively) is driven by the condensate hQTE�Qi = �2h �URUL + �DRDLi. The condensate is

namely invariant under the transformations Q ! gQ = exp(i�iT i)Q for the unbroken generators, i.e.

the SO(4) generators for E+ and Sp(4) for E�, which are shown in Appendix A. The transformation of

the condensate is

Q0TE�Q0 =QT gTE�gQ = QT (I + i�i(T i)T )E�(I + i�jT j)Q+O(�2)=QTE�Q+ iQT

��i(T i)TE� + �jE�T j

�Q+O(�2):

(3.14)

Thus, the condensate is invariant if the following criterion is satis�ed

TTi E� + E�Ti = 0: (3.15)

These kind of vacua are called technicolor vacua, because the chiral symmetry and the EW symmetry of

their condensate break simultaneously. In the composite-Higgs vacua which will be introduced in the next

chapter, the EW symmetry is unbroken after the chiral symmetry breaking. These vacua are discussed

for �rst time in Refs. [16, 17]. The representation of SU(4) in Eq. (7.1) in Appendix A can be inserted

into the criterion in Eq. (3.15) to show that the condensate is invariant under SO(4) for + sign and

Sp(4) for � sign, respectively. By inserting the Sa generators (Eqs. (7.222)-(7.223) in Appendix C-3),

we obtain

(Sa)TE� + E�Sa =

0@ �B� �B� 0

0 �B �B

1A ; a = 1; : : : ; 6; (3.16)

and by inserting the Xi generators, we get

(Xi)TE� + E�Xi =

0@ �D� +D� 2CT

�2C D �D

1A ; i = 1; : : : ; 9: (3.17)

Page 53 of 193


We have that B = 0 for the unbroken generators Sa when a = 1; : : : ; 4, A = 0 for Sa when a = 5; 6,

D = 0 for the broken generators Xi when i = 1; 2; 3, and C = 0 for Xi when i = 4; : : : ; 9. Therefore, for

E+ the relation in Eq. (3.15) is maintained for the generators Sa but not for Xi. For E� the relation is

maintained for the generators Sa where a = 1; : : : ; 4 and for Xi where i = 4; : : : ; 9, which are the Sp(4)

generators. Thus, the condensate in Eq. (3.12) is invariant under SO(4) transformations for E+ and

invariant under Sp(4) transformations for E�. Therefore, we use the vacuum matrix E+ in this theory,

because the SU(4) symmetry spontaneously breaks to SO(4) driven by the condensate hQTE+Qi. Thisleaves us with nine broken generators with associated Goldstone bosons.

In Eqs. (7.22)-(7.37) in Appendix B is shown that the gauge anomalies cancel in the SM. In Eqs.

(7.38)-(7.42) in Appendix B is shown that the gauge anomalies cancel in MWT, when we have following

generic hypercharge assignment

Y (QL) =y

2; Y (UR; DR) =

�y + 1

2;y � 1

2

�;

Y (LL) = �3y2; Y (NR; ER) =

��3y + 1

2;�3y � 1

2

�;

(3.18)

where the parameter y can be any real value for this theory, and the electric charge is Q = T3 + Y ,

where T3 is the weak isospin generator and Y is the hypercharge. If y = 1=3 then we recover the SM

hypercharge assignment.

By using the matrix notation of � in Eq. (7.96) in Appendix F and the left- and right-handed Dirac

spinor and their adjoint in Eqs. (7.216)-(7.219) in Appendix C-3, we can rewrite the following terms in

terms of Dirac spinors to in terms of Wayl spinors instead as follows

i �UL �D�UL + i �UR

�D�UR =i�

0 (UyL) _�;a�0@ 0 �� _�

�� _�� 0

1ADab

�

0@ UL;�;b

0

1A+

i�

(UyR)�;a 0

�0@ 0 �� _�

�� _�� 0

1ADab

�

0@ 0

U _�;bR

1A

=i�UyL�_�;a

�� _��Dab� UL;�;b + i

�UyR��;a

�� _�Dab� U

_�;bR :

By using this rewriting, the doublets and the singlets of the technifermions in Eq. (3.3) and (3.4), we

can rewrite the guage-kinetic Lagrangian in Eq. (3.9) to

LK = i �QL �D�QL + i �UR

�D�UR + i �DR �D�DR: (3.19)

The gauge-kinetic terms of the New Leptons have the same form as the techniquarks. Thus, we can

replace the Higgs sector of the SM with the MWT Lagrangian

LMWT =� 1

4F a��F

a�� + i �QL �D�QL + i �UR

�D�UR + i �DR �D�DR (3.20)

+ i�LL �D�LL + i �NR

�D�NR + i �ER �D�ER;

where the technicolor �eld strength tensor is F a�� = @�Aa��@�Aa�+gTC"abcAb�Ac� , the covariant derivative

Page 54 of 193


of the left-handed techniquarks is

Dab� = �ab@� + gTCA

c�"abc � ig ~W� � ~�

2�ab � ig0B�Y �ab; (3.21)

where the A� �elds are the techni gauge bosons, and W� and B� are the weak gauge bosons associated

to weak isospin SU(2)W and the hypercharge U(1)Y, respectively. The �a matrices are the Pauli matrices

(Eq. (7.3) in Appendix A), and "abc is the antisymmetric tensor. For right-handed technifermions the

third term in Eq. (3.21) containing the weak interactions disappears and for the New Leptons the second

term containing the technicolor interactions disappears. The hypercharge generator Y in the last term is

replaced with the appropriate hypercharge assignment in Eq. (3.18).

3.2 Low Energy Theory for MWT

In this section, we want to construct the e�ective theory for MWT, which includes the composite scalars

(e.g. a Higgs scalar), the composite vector bosons and the SM fermions. The e�ective theory will include

these composite particle self interactions and their interactions with the electroweak gauge �elds.

We will focus on bilinears, because we expect they dominate at low energy.1 If we have two spin-1/2

techniquarks, then we can construct scalar bilenears (spin-0) and vector bilinears (spin-1) from the Q

vector in Eq. (3.7) as follows

s = 0 : MAB � QA�;aQB;a� "�� ;

s = 1 : A1�;BA � Q�A;a�

�

� _�Q�

_�;B;a or

A2�;BA � Q�C;a�

�

� _�Q�

_�;C;a�BA with A;B = 1; : : : ; 4;

(3.22)

because 12 1

2 = 0 + 1. If we have one spin-1/2 techniquark and one spin-1 gauge boson, then we can

also construct a spin-1/2 bilinear as follows

s = 1=2 : PA; _� � �� _��QA�;aAa� with A;B = 1; : : : ; 4; (3.23)

because 12 1 = 1

2 +23 .

We start by describing the scalar sector, and thereafter we will describe the vector boson sector. In

the end of the section, we describe the fermions in the e�ective theory and their Yukawa coupling to the

scalar resonances.

3.2.1 Composite Scalars

We want to construct a relevant e�ective theory for the Higgs sector like in QCD but instead at the

electroweak scale. This e�ective theory consists of a composite Higgs and its pseudoscalar partner, and

nine pseudoscalar Goldstone bosons and their scalar partners. These composite particles are assembled

1The Goldstone bosons is among bilinears that dominating mostly, and for non-GBs we assume that their masses scalewith the number of the constituent fermions

Page 55 of 193


into a 4� 4 complex matrix M with the quantum numbers of the �rst techniquark bilinear in Eq. (3.22).

We can show that this techniquark bilinear is symmetric by switching the two Q vectors as follows

MAB � QA�;aQB�;b�

ab"�� = �QB�;bQA�;a�ab"�� = (�1)2QB�;bQA�;a�ba"�� = QB�;aQA�;b�

ab"�� MBA:

Therefore, we need to construct the matrix M such that it is symmetric by combining the broken

generators and the vacuum matrix E � E+. We get that

M =h�2+ ip2�aXa

iE; (3.24)

where the Xa's with a = 1; � � � ; 9 are the broken generators of the SU(4) group, which are listed in

Appendix A. The � = v+h �eld is a scalar which may acquire a vev v and the �a �elds are pseudoscalars.

This M matrix transforms under SU(4) group according to

M ! gMgT ; g 2 SU(4); (3.25)

where the SU(4) element can be expanded as g = exp(i�aT a) ' 1 + i�aT a with a = 1; : : : ; 15 for

in�nitesimal small �a phases.

We can make a SU(4) transformation of the M matrix

M !M 0 = gMgT =M + i�a[T aM +MT aT ] +O(�2); (3.26)

Thus, the M matrix is invariant under the SU(4) generators T a, if T aM +MT aT ' 0. The �rst term

of the M matrix is the only term that is invariant under SO(4) transformations, because the criteria

SaE + ESaT = 0 are maintained while XiE + EXiT 6= 0 according to Eq. (3.16) and Eq. (3.17).

Therefore, the broken generators do not leave the vacuum expectation value (VEV) of M invariant

hMi = v

2E: (3.27)

The second term of the M matrix is invariant under the SO(4) transformation but not invariant under

SU(4). This is shown in Eq. (7.224) in Appendix C-3, which gives that

SbXaE +XaESbT = 0:

Therefore, the M matrix in Eq. (3.24) is not a representation of SU(4). We can transform the M

matrix in Eq. (3.24) where it is written in terms of the �elds � and �a as follows

M !gMgT ' �1+ i�bT b� h�

2+ ip2�aXa

iE�1+ i�bT bT

�=

�� + i�(�aT a + E�aT aTE)

2+p2�i�a +

��cT c�a � �c�bXbT cTEXa��Xa

�E:

(3.28)

Therefore, we need to add an extra psudoscalar, �, and nine extra scalars, ~�a, to make the M matrix

closed under SU(4) transformations. Thus, we have that the M matrix is

Page 56 of 193


M =

�� + i�

2+p2(i�a + ~�a)Xa

�E: (3.29)

ThisM matrix is a representation of SU(4), which consists of 20 degrees of freedom or 10 complex degrees

of freedom. We have the �elds: �, �, �a and ~�a with a = 1; : : : ; 9.

The connection between the composite scalars in Eq. (3.29) and the underlying techniquarks can be

derived by observing that the elements of the matrixM transform like a techniquark bilinears as the �rst

bilinear in Eq. (3.22)

MAB � Q�AQ�B"�� with A;B = 1; : : : ; 4; (3.30)

By using this connection, the SU(4) generator matrices in Appendix A and the spinor bilinears in

Appendix F, we have related the scalar �elds to the wavefunctions of the techniquark bound states. The

results are shown in Eq. (7.97) and Eq. (7.98) in Appendix F. This gives for the technimesons, which

are composed of one techniquark and one anti-techniquark, the following charge states

v +H �� UU + �DD; � � i( �U 5U + �D 5D);

A0 �~�3 � �UU � �DD; �0 � �3 � i( �U 5U � �D 5D);

A+ �~�1 � i~�2

p2

� �DU; �+ � �1 � i�2

p2

� i �D 5U;

A� �~�1 + i~�2

p2

� �UD; �� 1 + i�2

p2

� i �U 5D:

(3.31)

For the technibaryons made up of two techniquarks (with two di�erent colors), we have that

�UU ��4 + i�5 +�6 + i�7

2� UTCU;

�DD ��4 + i�5 ��6 � i�7

2� DTCD;

�UD ��8 + i�9

p2

� UTCD;

~�UU �~�4 + i~�5 + ~�6 + i~�7

2� iUTC 5U;

~�DD �~�4 + i~�5 � ~�6 � i~�7

2� iDTC 5D;

~�UD �~�8 + i~�9

p2

� iUTC 5D;

(3.32)

where U = (UL;�; U_�R)

T and D = (DL;�; D_�R)T are the up- and down-techniquark, and C is the charge

conjugation matrix (shown in Eq. (7.96)). To these technibaryon charge states we have also their

corresponding charge conjugate states, e.g. instead of �UU we have �UU by making the substitution in

Eq. (7.105) in Appendix F. As shown in Appendix F, the elements of the M matrix can be rewritten in

terms of these technimeson and technibaryon charge states as follows

Page 57 of 193


M =

0BBBBBB@

i�UU + ~�UUi�UD+~�UDp

2�+i�+i�0+A0

2i�++A+p

2

i�UD+~�UDp2

i�DD + ~�DDi��+A�p

2�+i��i�0�A0

2

�+i�+i�0+A0

2i��+A�p

2i�UU + ~�UU

i�UD+~�UDp2

i�++A+p2

�+i��i�0�A0

2i�UD+

~�UDp2

i�DD + ~�DD

1CCCCCCA: (3.33)

The electroweak subgroup can be embedded in SU(4). The generators Sa with a = 1; 2; 3 in Appendix

A form the vectorial SU(2) subgroup of SU(4), denoted SU(2)V, and the generator S4 forms a U(1)V

subgroup. These two subgroups together with the broken generators Xa with a = 1; 2; 3 generate a

SU(2)L SU(2)R U(1)V subgroup of SU(4). This can be seen by changing generator basis (Sa; Xa) to

La � Sa +Xa

p2

=

0@ �a=2 0

0 0

1A ; �RaT � Sa �Xa

p2

=

0@ 0 0

0 ��aT =2

1A ;

S4 =1

2p2

0@ I 0

0 �I

1A ;

(3.34)

with a = 1; 2; 3. By gauging SU(2)L (identifying it with SU(2)W) and U(1)Y � SU(2)R U(1)V, the

electroweak gauge group SU(2)W U(1)Y is obtained, where

Y = �R3T +p2YV S

4; (3.35)

and YV is the U(1)V charge.From the general gauge anomaly free hypercharge assignment in Eq. (3.18),

we see that YV = y for the techniquarks, and YV = �3y for the New Leptons, because

Y QL;� =1

2

0BBBBBB@

YV UL;�

YVDL;�

�(YV + 1)UR;�

�(YV � 1)UR;�

1CCCCCCA

=1

2

0BBBBBB@

yUL;�

yDL;�

�(y + 1)UR;�

�(y � 1)DR;�

1CCCCCCA

and (3.36)

Y LL;� =1

2

0BBBBBB@

YVNL;�

YVDE;�

�(YV + 1)NR;�

�(YV � 1)ER;�

1CCCCCCA

=1

2

0BBBBBB@

�3yNL;��3yEL;�

�(�3y + 1)NR;�

�(�3y � 1)ER;�

1CCCCCCA: (3.37)

When SU(4) spontaneously breaks to SO(4), then the global subgroup SU(2)L SU(2)R breaks to

SU(2)V � SU(2)L+R as seen from Eq. (3.34) where the Xa are broken. The consequence is that the

electroweak gauge group breaks to U(1)Q, where

Q =p2S3 +

p2YV S

4: (3.38)

In summary, the global subgroup breaking pattern is SU(2)LSU(2)RU(1)V ! SU(2)VU(1)V (as in

two �avor QCD). The resulting EW symmetry breaking pattern is the coset SU(2)W U(1)Y ! U(1)Q.

The SU(2)V group acts as the custodial isospin as in the SM, which is entirely contained in the unbroken

Page 58 of 193


SO(4) group. This ensures that the � parameter in Eq. (2.72) is equal to one at tree-level.

The gauging of the electroweak symmetry breaks explicitly the SU(4) symmetry group down to

SU(2)L SU(2)R U(1)V (the gauging of SU(2)L gives SU(2)W and the gauging of the rest gives

U(1)Y � SU(2)RU(1)V), while the spontaneous symmetry breaking leaves a SO(4) subgroup invariant.

Therefore, the remaining unbroken group is SU(2)V U(1)V as simple illustrated in Figure 3.1. The

gauging of this group gives U(1)Q � SU(2)V U(1)V. Here is the U(1)Q group is the symmetry group

which is associated to the electromagnetism, while the U(1)V symmetry leads to the conservation of the

technibaryon number.

SU(2)L � SU(2)R � U(1)V SU(2)V � U(1)V

SU(4)

SO(4)

Figure 3.1: Spontaneous breaking from SU(4) to SO(4) due to dynamics and explicit breaking from SU(4)to SU(2)L SU(2)R U(1)V due to EW gauging.

By using Eq. (3.38), we can calculate the charges of the technimesons and technibaryons. Firstly,

we will �nd the charges of the elements of the Q vector in Eq. (3.7). In this case we have that YV = y

because the charge operator Q works on techniquarks. We have that

QQAL;� =�p

2S4 +p2YV S

4�QL;� =

1

2

0BBBBBB@

(1 + y)UL;�

(�1 + y)DL;�

(�1� y)UR;�(1� y)DR;�

1CCCCCCA: (3.39)

By adding the two charges for the two techniquarks we get the charges of the elements of the M matrix,

which are

MAB � Q�L;AQL;B;� ) QAB =

0BBBBBB@

1 + y y 0 1

y �1 + y �1 0

0 �1 �1� y �y1 0 �y 1� y

1CCCCCCA

with A;B = 1; : : : ; 4: (3.40)

In Table 3.2 the scalars are classi�ed according to the unbroken group U(1)VU(1)Q with the U(1)V

charge and the U(1)Q charge, which are illustrated as the unbroken symmetry group in Figure 3.1. Three

of the nine physical degrees of freedom are eaten up by the longitudinal components of the SM gauge

Page 59 of 193


bosons, while the remaining six Goldstone bosons carry technibaryon number which are denoted by �UU ,

�DD, �UD and their charge conjugated states. Because these GBs carry technibaryon number, we refer

to these states as technibaryons.

Field U(1)V charge U(1)Q charge Linear Combination

W+L / �+ 0 +1 �1�i�2p

2

W�L / �� 0 -1 �1+i�2p

2

ZL / �0 0 0 �3

�UU / ~�UU +1 y + 1 �4+i�5+�6+i�7

2 /~�4+i~�5+~�6+i~�7

2

�DD / ~�DD +1 �1 + y �4+i�5��6�i�7

2 /~�4+i~�5�~�6�i~�7

2

�UD / ~�UD +1 y �8+i�9p2

/~�8+i~�9p

2

�yUU / ~�yUU -1 �y � 1 �4�i�5+�6�i�7

2 /~�4�i~�5+~�6�i~�7

2

�yDD / ~�yDD -1 1� y �4�i�5��6+i�7

2 /~�4�i~�5�~�6+i~�7

2

�yUD / ~�yUD -1 �y �8�i�9p2

/~�8�i~�9p

2

Table 3.2: Classi�cation of the Goldstone bosons according to the unbroken global group U(1)V and theunbroken gauge group U(1)Q � SU(2)V U(1)V.

In the following we will show that the �elds A0;� and �0;� are triplets under the custodial symmetry

SU(2)V, while the �elds � and � are singlets. We know from Eq. (3.33) that the M matrix can be

written in the symmetric form

M =

0@ A B

BT C

1A ; (3.41)

where A, B and C are 2� 2 matrices. We have the elements of the SU(2)L and SU(2)R are written as

g4�4L =ei�aLa =

0@ ei�

a�a 0

0 0

1A =

0@ gL 0

0 0

1A ;

(g4�4R )� =e�i�aRaT =

0@ 0 0

0 e�i�a�aT

1A =

0@ 0 0

0 g�R

1A ;

(3.42)

where La and �RaT are the generators in Eq. (3.34), and g�R = (exp(i�a�a))� = exp(�i�a�a�) =

exp(�i�a�aT ). Thus, the M matrix transforms under the SU(2)L SU(2)R symmetry group as

M ! gMgT =

0@ gLAg

TL gLBg

yR

g�RBT gTL g�RCg

yR

1A ; (3.43)

where

g4�4 = g4�4L + (g4�4R )� =

0@ gL 0

0 g�R

1A : (3.44)

The B matrix can be written in the form

Page 60 of 193


B =� + i�

21+

~�i + i�i

2� i: (3.45)

The chiral symmetry breaking SU(4) ! SO(4) gives rise to that the group SU(2)L SU(2)R U(1)V

breaks to SU(2)VU(1)V, where SU(2)V is the custodial symmetry group, i.e. gL = gR = gV . Thus, the

�rst term in B is invariant under SU(2)V transformations as follows

B(1) ! gVB(1)gyV =

�1+ i�a�a

�� + i�

21�1� i�b� b�+O(�2)

=� + i�

21+O(�2);

(3.46)

while the second term in B transforms as follows

B(2) ! gVB(2)gyV =

�1+ i�a�a

� ~�i + i�i

2� i�1� i�b� b�+O(�2)

=~�i + i�i

2� i + i�a

~�i + i�i

2[�a; � i] +O(�2):

(3.47)

The �elds �i and ~�i mix with each other by transforming them under SU(2)V, respectively. Thus, the

�elds A0;� and �0;� in Eq. (3.31) form each a triplet under SU(2)V. However, the �elds � and � are

both a singlet under SU(2)V.

We will now construct an e�ective Lagrangian with the M matrix as in QCD. The electroweak

covariant derivative for the M matrix has the form

D�M = @�M � ig[G�(YV )M +MGT� (YV )]; (3.48)

where we have YV = y because we have to take the U(1)Y charge of the techniquark constituents in the

M matrix as shown in Eq. (3.30), and

gG�(YV ) =gWa�L

a + g0B�Y

=gW a�L

a + g0B��R3T +

p2YV S

4�:

(3.49)

Under electroweak gauge transformations, we have that M transforms as follows

M(x)! u(x; y)M(x)uT (x; y); (3.50)

where

u(x;YV ) = exphi�a(x)La + i�(x)

��R3T +

p2YV S

4�i; (3.51)

and YV = y because the M matrix consists of techniquarks.

We can construct an e�ective Lagrangian at low energy. The e�ective Lagrangian must respect the

global symmetries as the underlying Lagrangian. Furthermore, it must be invariant under the electroweak

gauge transformations and CP transformations. The new Higgs Lagrangian is

LHiggs = 1

2Tr[D�MD�My]� V (M) + LETC; (3.52)

where the potential is

Page 61 of 193


V (M) = �m2

2Tr[MMy] +

�

4Tr[MMy]2 + �0Tr[MMyMMy]� 2�00[DetM +DetMy]; (3.53)

and LETC is all the terms which are generated by the extended technicolor interactions (ETC) and not by

the chiral symmetry breaking sector. We can not use the counting scheme in derivatives as in Eq. (2.130)

for the �rst three terms in the potential, but as discussed below Eq. (2.132) we can ignore many-particle

vertices. Firstly, it is hard to produce and therefore not interesting to consider. Secondly, at a given

number of external lines in the vertex then the energy would be above the scale where the composite

particles would fall apart.

All the terms in the Lagrangian LHiggs are invariant under a global transformations, gauge transfor-

mations and CP transformations. In Appendix G the discrete transformations (parity, charge conjugation

and CP transformations) of spinors, the Q vector and the M matrix are derived. We can notice that the

derterminant terms explicitly break the U(1)A symmetry, which give mass to �. This excitation would

otherwise be a massless Goldstone boson.

Three of the nine Goldstone bosons associated with the nine broken generators Sa become longitudinal

degrees of freedom of the massive weak gauge bosons. The last six Goldstone bosons will achieve a mass

from the extended technicolor interactions (ETC) and the electroweak interactions. According to Eq.

(26) in Ref. [1], the ETC interaction terms can be written as follows

LETC =m2ETC

4Tr[MBMyB +MMy] + � � � ; (3.54)

where B � 2p2S4 and the extra terms could be higher dimensional operators. ETC terms generate also

the masses of the SM fermions as explained later.

Finally, we can determine from the Lagrangian in Eq. (3.52) the vacuum expectation value (VEV)

of the composite Higgs and the masses of the composite scalars in terms of the model parameters. By

using the Mathematica, the vacuum expectation value (VEV) of the Higgs candidate is

v2 = h�i2 = m2

�+ �0 + �00: (3.55)

By using the Mathematica, we have that the Higgs mass term and therefore the Higgs mass is

� M2H

2H2 =

�1

2m2 � 3

2v2�� 3

2v2�0 +

3

2v2�00

�H2 =

�1

2m2 � 3

2m2

�H2 = �m2H2

)M2H = 2m2: (3.56)

The same procedure is carried out for the remaining composite technimesons and technibaryons. The

masses of the remaining technimesons are

�M2�

2�2 =

�m2

2� 1

2v2�� 1

2v2�0 � 3

2v2�00

��2 =

�1

2(�+ �0 � �00)� 1

2v2�� 1

2v2�0 � 3

2v2�00

��2

=� 2v2�00�2 )M2� = 4v2�00; (3.57)

Page 62 of 193


which reveives its mass from the explicitly breaking of the U(1)A symmetry by the determinant terms

with the coupling constant �00 in Eq. (3.53), and

� 1

2M2A�

�A+A� +A�A+

�= �M

2A�

2

�~�21 +

~�22

�= �(v2�0 + v2�00)

�~�21 +

~�22

�)M2

A� = 2v2(�0 + �00); (3.58)

�1

2M2A0A0A0 = �M

2A0

2~�3 ~�3 = �v2(�0 + �00)~�3 ~�3 )M2

A0 = 2v2(�0 + �0;0) (3.59)

and the three pseudoscalar mesons �� and �0 are massless, and they correspond to the three massless

Goldstone bosons which are eaten by the longitudinal degrees of freedom of the massiveW� and Z boson.

The remaining six uneaten Goldstone bosons are the technibaryons, which acquire tree-level degenerate

masses by the not speci�ed ETC interactions

� 1

2M2

�UU��UU�UU �

1

2M2

�DD��DD�DD = �1

4

�M2

�UU

2+M2

�DD

2

�(�2

4 +�25 +�2

6 +�27)

= �1

4m2ETC(�

24 +�2

5 +�26 +�2

7))M2�UU =M2

�DD = m2ETC and (3.60)

� 1

2M2

�UD��UD�UD = �M

2�UD

2(�8 � i�9)(�8 + i�9) = �

M2�UD

2(�2

8 +�29) = �

m2ETC

2(�2

8 +�29)

)M2�UD = m2

ETC ; (3.61)

The degenerate mass of the remaining technibaryons is

� 1

2M2

~�UU

�~� �U �U

~�UU + ~�UU ~� �U �U

�� 1

2M2

~�DD

�~� �D �D

~�DD + ~�DD ~� �D �D

�= �

�M2~�UU

2+M2

~�DD

2

�(~�2

4 +~�25 +

~�26 +

~�27)

= �m2ETC

2(~�2

4 +~�25 +

~�26 +

~�27)� v2(�0 + �00)(~�2

4 +~�25 +

~�26 +

~�27)

)M2~�UU

=M2~�DD

= m2ETC + 2v2(�0 + �00);

(3.62)

and

� 1

2M~�UD

(~� �U �D~� �U �D + ~�UD ~� �U �D) = �

M2~�UD

2(~�8 � i~�9)(~�8 + i~�9) = �

M2~�UD

2

�(~�8)2 + (~�9)2

�= �m

2ETC

2

�(~�8)2 + (~�9)2

�� v2(�0 + �00)�(~�8)2 + (~�9)2

�)M2

~�UD= m2

ETC + 2v2(�0 + �00): (3.63)

3.2.2 Composite Vector Bosons

The composite vector bosons of this theory are conveniently described by

A� = Aa�T a; (3.64)

where T a are the SU(4) generators with T a = Sa for a = 1; : : : ; 6, and T a+6 = Xa for a = 1; : : : ; 9 in

Page 63 of 193


Appendix A. We have that A� transforms under an SU(4) transformation as follows

A� ! gA�gy; where g 2 SU(4): (3.65)

According to the tracelessness of the matrix A� in Eq. (3.64) and the SU(4) transformation of the matrix

in Eq. (3.65), this gives a connection of this matrix A� with the two lower techniquark bilinears in Eq.

(3.22)

A�;BA � Q�A��

� _�Qy

_�;B � 1

4Q�C�

�

� _�Qy

_�;C�BA with A;B;C = 1; : : : ; 4; (3.66)

which is traceless because Tr(A�;BA ) = QA��QyA � 1

4�AAQC�

�QyC = 0, and it transforms as Eq. (3.65),

because

A�;BA = QA��QyB � 1

4QC��QyC�BA �!

gCAQC��QyDgyBD � 1

4gDCQD�

�QyEgyCE �BA = gCAQC��QyDgyBD � 1

4QE��QyE�BA =

gCAQC��QyDgyBD � 1

4gCAQE�

�QyE�DC gyBD = gCAA

�;DC gyBD :

(3.67)

In Appendix F, the relations between the charge eigenstates and the wavefunctions of the composite

vector mesons are derived, which are

v0� �A3� � �U �U � �D �D; a0� � A9� � �U � 5U � �D � 5D;

v+� �A1� � iA2�

p2

� �D �U; a+� � A7� � iA8�

p2

� �D � 5U;

v�� A1� + iA2�

p2

� �U �D; a�� A7� + iA8�

p2

� �U � 5D;

v4� �A4� � �U �U + �D �D;

(3.68)

and for the vector baryons we have that

x�UU �A10� + iA11� +A12� + iA13�

2� UTC � 5U;

x�DD �A10� + iA11� �A12� � iA13�

2� DTC � 5D;

x�UD �A14� + iA15�

p2

� DTC � 5U;

s�UD �A6� � iA5�

p2

� UTC �D:

(3.69)

In Eq. (7.113) in Appendix F, we have also derived the A� matrix which is de�ned in Eq. (3.64) with

the vector technimesons and technibaryons in Eq. (7.106) and (7.107)

A� =

0BBBBBB@

a0�+v0�+v4�

2p2

a+�+v+�

2

x�UUp2

x�UD

+s�UD

2

a��+v��

2�a0��v0�+v4�

2p2

x�UD

�s�UD

2

x�DDp2

x�UUp2

x�UD

�s�UD

2a0��v0��v4�

2p2

a��v��2

x�UD

+s�UD

2

x�DDp2

a+��v+�2

�a0�+v0��v4�2p2

1CCCCCCA: (3.70)

Page 64 of 193


The kinetic Lagrangian is

Lkin =� 1

2Tr[ ~W��

~W�� ]� 1

4B��B

�� 1

2Tr[F��F

�� ] +m2ATr[C�C

�]; (3.71)

where ~W�� and B�� are the ordinary �eld strength tensors for the electroweak gauge �elds and F�� is

the new �eld strength tensor for the new SU(4) vector bosons, which is

F�� = @�A� � @�A� � i~g[A�; A� ]; (3.72)

and we have de�ned the vector �eld C� as follows

C� � A� � g

~gG�(y); (3.73)

where G�(y) is the vector �eld in Eq. (3.49) with YV = y. The tensor ~W�� are not yet the SM weak

triplets. They mix with the composite vector bosons to form mass eigenstates which corresponding to

the ordinary 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;YV ) is given by Eq. (3.51). This vector �eld transform like a gauge �eld with the exception

of the extra term with u@�uy in gauge transformations.

The terms in the Lagrangian are not only kinetic ones, because it contains self-interaction terms and

one mass term. The mass term is gauge invariant, which gives a degenerate mass mA to all the composite

bosons, while leaving the gauge bosons massless. The gauge bosons acquire their mass from the covariant

derivative term of the scalar matrix M in Eq. (3.52) after spontaneous symmetry breaking.

We can construct an e�ective Lagrangian where the C� �elds couple to theM matrix up to dimension

four operators. The e�ective Lagrangian can be written as

LM�C =~g2r1Tr[C�C�MMy] + ~g2r2Tr[C�MC�TMy] + i~gr3Tr

�C�(M(D�M)y � (D�M)My)

�+

~g2sTr[C�C�]Tr[MMy];

(3.75)

where the dimensionless parameters r1, r2, r3 and s are the di�erent strength of the interactions between

the composite scalars and vectors in units of ~g, therefore they are expected to be of order one. The terms

in the e�ective Lagrangian are global SU(4) invariant, gauge invariant and CP invariant.

3.2.3 Fermions in the E�ective Theory

The fermionic content of the e�ective theory consists of the SM quarks and leptons, a composite techniquark-

technigluon doublet, and the New Lepton doublet which is introduced to cure the Witten anomaly.

We want to extend the SU(4) symmetry to the ordinary quarks and leptons. We arrange the SU(2)W

doublets in SU(4) multiplets as we have done for the techniquarks in Eq. (3.7). For the SM quarks and

leptons, we introduce the four component vectors

Page 65 of 193


qA;iL;� =

0BBBBBB@

uiL;�

diL;�

"��(u�R)i;�

"��(d�R)i;�

1CCCCCCA

and lA;iL;� =

0BBBBBB@

�iL;�

eiL;�

"��(��R)i;�

"��(e�R)i;�

1CCCCCCA; (3.76)

where i is the generation index. To have this extended SU(4) symmetry then we need to introduce a

right-handed neutrino for each generation. In addition to these SM SU(4) multiplets, we have an multiplet

for the New Leptons and techniquark-technigluon bound state,

LAL;� =

0BBBBBB@

NL;�

EL;�

"��(N�R)�

"��(E�R)�

1CCCCCCA

and ~QAL;� = QyA;_�

L �� _�A� =

0BBBBBB@

~UL;�

~DL;�

"��( ~U�R)�

"��( ~D�R)�

1CCCCCCA: (3.77)

We can write the fermion Lagrangian with a SU(4) global symmetry as follows

Lfermion =iqyi_� �� _��D�qi� + ilyi_� ��

� _��D�li� + iLy_��

� _��D�L� + i ~Qy_�� _��D�

~Q�+

x ~Qy_�� _��C� ~Q� ;

(3.78)

where the electroweak covariant derivative for the fermion �elds can be written as

D� = @� � igG�(YV ); (3.79)

where G�(YV ) is given in Eq. (3.49), and the vector �eld C� is de�ned in Eq. (3.74). The U(1)V charge

is YV = 1=3 for the SM quarks, YV = �1 for the SM leptons, YV = �3y for the New Lepton doublet,

and YV = y for the techniquark-technigluon bound state. The �rst four terms in the Lagrangian are the

kinetic terms of the fermions like the Lagrangian term in Eq. (3.10).

The last term in the Lagrangian which couples ~Q to C� is always allowed, because the term is invariant

under electroweak gauge transformations for any YV = y. Any SU(4) fermion multiplet transforms as

follows

(x)! u(x;YV ) (x); (3.80)

and C� transforms as follows

C�(x)! u(x; y)C�(x)u(x; y)y; (3.81)

where u(x;YV ) is given in Eq. (3.51). We have that YV = y for C� due to the fact that the composite

vectors are built out of techniquark bilinears. Thus, we have that the term y�� _��C� transforms like

y�� _��C� �! yu(x;YV )y�� _��u(x; y)C�u(x; y)yu(x;YV ) ; (3.82)

and therefore the term is only invariant if YV = y. For y 6= 1=3 and y 6= �1, we have that the term is

only invariant for = ~Q (the last term in Eq. (3.78)). For y = 1=3 or y = �1, we have that the term is

not only invariant for = ~Q, but also for either = qi or = li, respectively.

Page 66 of 193


3.2.4 Yukawa Interactions

In this section we will provide masses to ordinary fermions. There are many extensions of technicolor

to provide the fermion masses. One way could be to use another strongly coupled gauge dynamics or

introduce new fundamental scalars. Such a model is called an extended technicolor (ETC) theory, which

we discuss later.

In this section we are simply couple the fermions to our low energy e�ective Higgs to keep the number

of �elds minimal. This is done by writing Yukawa interactions which couple the SM fermions to the

matrix M . These Yukawa terms are depending on the value of y for the techniquarks. We denote as

either qi or li. We can write the Yukawa term

� TM� + h.c.; (3.83)

which is electroweak gauge invariant, when the U(1)V charges of and the techniquark multiplets Qa

are the same. The Yukawa term is invariant for YV = y, because it transforms (according to Eq. (3.50)

and (3.80)) as follows

� TM� �! � Tu(x;YV )Tu(x; y)�M�u(x; y)yu(x;YV ) ; (3.84)

where u(x; y)yu(x; y) = u(x; y)u(x; y)y = 1, and therefore we have that u(x; y)Tu(x; y)� = 1. Otherwise,

if the U(1)V charges of and Qa are di�erent, then we can only write a gauge invariant Yukawa term

with the o�-diagonal M (contains only the Higgs boson and the Goldstone bosons), i.e.

Mo� �

0BBBBBB@

0 0 �+i�+i�0+A0

2i�++A+p

2

0 0 i��+A�p2

�+i��i�0�A0

2

�+i�+i�0+A0

2i��+A�p

20 0

i�++A+p2

�+i��i�0�A0

2 0 0

1CCCCCCA: (3.85)

This Yukawa term is written as

� TM�o� + h.c.; (3.86)

because the U(1)V charge of the Mo� is zero, since

S4Mo� +Mo�S4T = 0 (3.87)

according to Eq. (7.225) in Appendix C-3. Therefore, the U(1)V charges of T and need to cancel

each other in Eq. (3.86). The Yukawa term in Eq. (3.86) is the only viable for the New Leptons,

because the corresponding U(1)V charge is di�erent from the charge of the techniquark multiplets Qa

(YV = �3y 6= y). For the SM quarks, the Eq. (3.83) contains quark-quark terms which are not color

singlets. Therefore, the only viable Yukawa term for the ordinary quarks is the term in Eq. (3.86).

However, we notice that the Yukawa terms in Eq. (3.83) and (3.86) are not phenomenologically viable

yet, because the SU(2)L subgroup of SU(4) are unbroken and there are no distinguish between the up-type

and down-type fermions in these Yukawa terms. Therefore, we break the SU(2)L symmetry to U(1)R by

Page 67 of 193


using the projection operators (as done in eq. (2.77) when we talked about custodial symmetry)

PU =

0@ 1 0

0 1+�3

2

1A and PD =

0@ 1 0

0 1��32

1A (3.88)

Thus, we replace Eq. (3.83) and Eq. (3.86) with

T (PUM�PU ) � T (PDM�PD) + h.c.; (3.89)

and

T (PUM�o�PU ) � T (PDM�

o�PD) + h.c.: (3.90)

In the next, we would write the Yukawa interactions for two di�erent cases, y = �1 and y 6= �1.For y = �1, we can form gauge invariant Yukawa terms with the SM leptons and the full M matrix.

Therefore, the Yukawa Lagrangian for this case is

LYukawa =� yiju qiT (PUM�o�PU )q

j � yijd qiT (PDM�o�PD)q

j

� yij� liT (PUM�PU )lj � yije liT (PDM�PD)lj

� yNLT (PUM�o�PU )L� yELT (PDM�

o�PD)L

� y ~U ~QT (PUM�PU ) ~Q� y ~D ~QT (PDM

�PD) ~Q+ h.c.;

(3.91)

where yiju , yijd , y

ij� and yije are arbitrary complex matrices, and yN , yE , y ~U and y ~D are complex numbers.

For y 6= �1, we can only form gauge invariant Yukawa terms with the SM fermions and the o�-diagonal

M matrix

LYukawa =� yiju qiT (PUM�o�PU )q

j � yijd qiT (PDM�o�PD)q

j

� yij� liT (PUM�o�PU )l

j � yije liT (PDM�o�PD)l

j

� yNLT (PUM�o�PU )L� yELT (PDM�

o�PD)L

� y ~U ~QT (PUM�PU ) ~Q� y ~D ~QT (PDM

�PD) ~Q+ h.c.:

(3.92)

3.3 Extended Technicolor Models

In a technicolor model we need to incorporate a mechanism that generates quark and lepton masses, the

various weak mixing angles, and the CP-violation. Thus, we have that the quarks and leptons of the SM

need to couple to the techniquark condensate. In addition, there must be a mechanism that violates the

technibaryon quantum number, because the techniquarks must be able to decay, since there are no stable

technibaryons observed in the universe.

A popular way to solve these requirements is to extend the Technicolor gauge interactions with some

extended gauge bosons, which couple both to SM fermions and techniquarks. These extended interactions

are part of a large gauge group GETC which breaks down to the technicolor subgroup at an energy �ETC.

This energy scale �ETC is above the scale �TC at which the technicolor coupling becomes strong.

From a high-energy theory based on a master gauge group GETC, it is possible to obtain a low-energy

Page 68 of 193


theory where the only surviving gauge groups are those of technicolor and the SM. The master gauge

group GETC undergoes a symmetry breaking at the scale �ETC, where it breaks down to the technicolor

gauge group GTC as follows

GETC ! GTC GSM at �ETC; (3.93)

where the remaining groups in addition to GTC are include the full Standard Model GSM = SU(3)C SU(2)WU(1)Y. In the new interactions are required couplings of techniquarks QL;R into the SM quarks

and leptons L;R (qL;R and lL;R) with the currents of the form QL;R � L;R, which couple to the new

ETC gauge bosons. The full theory with the master gauge group GETC contains the desired currents of

the form � , Q � and Q �Q. A simple example could be that the Technicolor group SU(NTC) is

embedded into a larger ETC group SU(NETC), where of course we have that NETC > NTC.

At low energy scale � . �ETC, we have that the heavy ETC bosons, which exchange from the currents

corresponding to the broken ETC generators T a, produces three types of e�ective contact interactions

between the techniquarks and the SM fermions, which (cf. page 59 in Ref. [15]) are

�abQLT

aQR RTb L

�2ETC

+ �abQT aQQT bQ

�2ETC

+ ab LT

a R RTb L

�2ETC

+ : : : ; (3.94)

where the �ab, �ab and ab are coe�cents that are contracted with generator indices, where their structure

depends upon the construction of the ETC theory.

gLZ0�QL

�qL + gRZ0�QR

�qR

QL

qL

qR

QR

Q�L

qL q�R

QR

Z 0�gL gR

gLgRM2

Z0qLQ

�LQRq

�R

1M2

Z0�p2

gLgRM2

Z0

qL

q�R

Mo�gLgRv2EW

M2

Z0qLMo�q

�R

�q

��MZ0

�� 4�fTC�

�ETC �MZ0

vEW � hQQi1=3

Energy Scale

Figure 3.2: The various symmetry breakings from a ETC gauge symmetry GETC for ETC gauge bosonsZ 0� to GSM after EW spontaneous symmetry breaking, which produce the masses of the SM fermions.

Page 69 of 193


The �-term in Eq. (3.94) is responsible for giving masses for the SM fermions

mf � hQLQRiTC�2ETC

= gLgRhQLQRiTCM2

ETC

; (3.95)

where gL and gR are the ETC gauge couplings to the left- and right-handed fermions, respectively,

METC = �ETC=pgLgR is the mass of the ETC gauge boson, and hQLQRiTC is the techniquark condensate

evaluated at the TC scale �TC. An illustration of the various symmetry breakings of an ETC theory

is shown in Figure 3.2, where the TC and EW breaking happen at same energy scale �TC � vEW

as in the SO(4) Minimal Walking Technicolor (MWT). At energies over �ETC the ETC gauge bosons

interact with both the technifermions and the fermions with the ETC gauge couplings gL;R. When the

energy is lowered below �ETC, then the ETC gauge propagator can be integrated out such that we

have an e�ective four fermion vertex. Finally, when the energy is lowered below the TC scale, then the

techniquarks condense and we get the condensate hQLQRiTC. In this special case, we have that the

condensate is hQLQRiTC = 4�v3EW. Overall, we have the following symmetry breaking pattern:

gLZ0�QL

�qL + gRZ0�QR

�qR��MZ0��! 1

�2ETC

qLQ�LQRq

�R =

gLgRM2Z0qLQ

�LQRq

�R

��vEW��! 4�f3

�2ETC

qLq�R =

gLgRv3EW

M2Z0

qLq�R;

(3.96)

where the Yukawa couplings are

�q =gLgRv

2EW

M2Z0

: (3.97)

The form of the matrix Mo� is shown in Eq. (3.85), where � = vEW + h. Therefore for this special case

the masses of the SM fermions in MWT are

mf � gLgRv3EWM2Z0; (3.98)

This mass formula can easily be generalized to the Eq. (3.95).

The fermion masses can also be produced by new heavy scalar �elds H which interact with the tech-

niquark and the SM fermions with the Yukawa coupling �Q and �q, respectively. The various symmetry

breakings are shown in Figure 3.3, where the new scalar propagators are integrated out, when the energy

is lowered below the ETC scale, which give the �-term in Eq. (3.94. When the energy is lowered below

TC scale, we get the Yukawa terms shown in the �gure, and thus the Yukawa terms in either Eq. (3.91)

or Eq. (3.92). Overall, we have the following symmetry breakings for new heavy scalars:

�QQLHQR + �RqLHqRE�MH��! 1

�2ETC

Q�LQRq�LqR =

�Q�qM2H

Q�LQRq�LqR

E�v��! 4�f3

�2ETC

q�LqR =�Q�qv

3EW

M2H

q�LqR;(3.99)

where the Yukawa couplings are

�q =�Q�qv

2EW

M2H

: (3.100)

Therefore for this special case with new heavy scalars the masses of the SM fermions in MWT are

Page 70 of 193


mf � �Q�qv3EW

M2H

: (3.101)

By extending the Technicolor theory by adding heavy gauge bosons or new heavy scalars, we have

moved the naturalness problem further up the energy scale, because we have a new scalar which mass

should be �ne-tuned. Therefore, we have only reduced the naturalness problem and not removed it. It is

also the case in the ETC theories with new gauge bosons, because we need new scalar �elds like the Higgs

boson in SM to make the gauge bosons massive after spontaneous symmetry breaking. Additionally, the

�-term contributes also to mixing angles between quarks and leptons, i.e. it contributes to the parameters

of the CKM and the PMNS matrix.

�QQLHQR + �qqLHqR

QL

QR

qL

qR

Q�L

QR q�L

qR

H�Q �q

�Q�qM2H

Q�LQRq�LqR

1M2H�p2

�Q�qM2H

qR

q�L

Mo��Q�qv2EW

M2H

q�LMo�qR�q

��MH

�� 4�fTC�

�ETC �MH

vEW � hQQi1=3

Energy Scale

Figure 3.3: The various symmetry breakings from a ETC gauge symmetry GETC for heavy scalars H toGSM after EW spontaneous symmetry breaking, which produce the masses of the SM fermions.

The mass hierarchy between the generations of the fermions can be achieved breaking GETC in several

steps as follows

GETC ! Gn ! Gn�1 ! � � � ! G1 ! GTC GSM : (3.102)

Some of the ETC gauge bosons become massive during the every step, which gives di�erent ETC scales

�ETC �METC. Thus, this scenario produces di�erent fermion masses as desired according to Eq. (3.95).

This way to produce the fermion mass hierarchy is called tumbling.

The �-term in Eq. (3.94) can induce masses to the pseudo-Goldstone bosons (pNGBs). The upper

diagram in Figure 3.4 shows how the ETC propagator is integrated out for energies � � �ETC, such

Page 71 of 193


that we obtain the four-technifermion operators. These four-technifermion terms can potentially solve a

problem that the masses of the PNGBs are too small that we have not observed them. This mechanism can

elevate the masses of these light PNGBs to larger values which are more consistent with the experiments.

For example in the SO(4) MWT the six pNGBs (the pNGBs which are not become the longitudinal

degrees of freedom of the weak gauge bosons) achieve their masses from these �-terms in Eq. (3.54). The

ETC terms in Eq. (3.54) consist both of two M matrices, i.e. these terms are four technifermion vertices

as the �-terms. Thus, their masses are M2�UU

= M2�DD

= M2�UD

= m2ETC and the same mass for their

charge conjugated �elds.

Finally, the -term in Eq. (3.94) generates Flavor-changing neutral current (FCNC) contributions

which exclude the possibility of generating large fermion masses in these ETC models. The lower diagram

in Figure 3.4 shows how the ETC propagator is integrated out for energies �� ETC, such that we obtain

the four-fermion operators.

QL

QR

qL

qR

�Q �q1

M2H�p2

��MH

Q�L

QR Q�L

QR

� �2QM2H

qL

qR

qL

qR

�q �q1

M2H�p2

��MH

q�L

qR q�L

qR

� �2qM2H

H

H

Figure 3.4: The upper diagrams are the ETC symmetry breaking which gives a �-term where the heavyscalar H propagator is integrated out. The lower diagrams are the ETC symmetry breaking which givesa -term.

For example a process like(�s 5d)(�s 5d)

�2ETC

(3.103)

is induced. This new contribution causes �S = 2 FCNC interactions which give a contribution to the

well-measured KLKS mass di�erence (short-lived KS (CP = �1) and long-lived KL (CP = +1) weak

eigenstate). This is an indirect way to measure of CP violation due to the mixing of the neutral kaons

K0 and its antiparticle �K0, because the K0 and �K0 has the quark content �sd and �ds, respectively, then

the four-fermion term in Eq. (3.103) contributes to this mixing and thus the CP violation. This -term

yields the contribution to the mass di�erence (according to Eq. (3.98) in Ref. [15])

�m2

m2K

� f2Km

2K

�2ETC

. 10�14; (3.104)

where fK is the kaon decay constant, mK is the kaon mass, and we expect that � sin2 �C � 10�2 in a

realistic model. Therefore, we obtain the lower constraint on the ETC scale

�ETC & 103 TeV: (3.105)

Page 72 of 193


where fK � 100 MeV and mK � 500 MeV. Applying this bound and assuming � � � � yields an

upper bound on the masses of the SM fermions, which is

mf . 100 MeV: (3.106)

Thus, it is already problematic to produce the mass of the charm quark with this ETC model. This

problem can maybe be alleviated by the coupling of the technicolor model is walking in an energy

window as explained in the next section.

3.4 Walking Technicolor

There are problems in building models with fermions there are heavy enough and models with su�ciently

suppressed �avor-changing neutral currents (FCNCs). The ETC models in previous section produce not

the observed quark and lepton masses. In this section we attempt to deal with these di�culties with

walking technicolor.

The Lagrangian of such a theory has the form

L = �Q �D�Q� 1

4Tr[G��G

�� ]; (3.107)

whereQ are the techniquarks andG�� is the �eld strength tensor of the technigluons. Let the techniquarks

be in the fundamental representation of SU(N) as the quarks in QCD. The last term of the Lagrangian

is the Yang-Mills theory, such a theory consists not of quarks. There is still con�nement in such a

theory, because at low energies there can be created glueballs, which is a hypothetical composite particle

consisting solely of gluon particles.

The �-function for the coupling g (from Eq. (2.6.34) in Ref. [13]) is

�(g) � @g

@ log�= �0

g3

(4�)2+ �1

g5

(4�)4+ �2

g7

(4�)6+O(g9); (3.108)

where

�0 =4

3TR � 11

3CA;

�1 =� 34

3C2A +

20

3CATR + 4CRTR;

�2 =� 2857

54C3A +

1415

27C2ATR �

158

27CAT

2R +

205

9CACRTR � 44

9CRT

2R � 2C2

RTR:

(3.109)

The Casimir operators CA, CR, and the Dynkin index TR are de�ned as follows

Xa;b

fabcfabd = CA�cd;

N2�1Xa=1

Xj

T aijTajk = CR�ik;

Tr(T aT b) = TR�ab;

(3.110)

Page 73 of 193


respectively. For the fundamental representation of SU(NC), we have that CA = NC , CR = CF =

(N2C � 1)=2NC and TR = TFNf = Nf=2, where Nf and NC are the number of fermions and colors,

respectively.

Nf

NC

�0 > 0, �1 > 0

�0 < 0, �1 > 0, �2 > 0

g

�

�0 < 0, �1 > 0, �2 < 0

�0 < 0, �1 < 0

g

�g

�

g

�

g

�

�

g

�

g

�

g

�

g

�

g

Banks-Zaks FP

QED-like

QCD-like

Walking Technicolor

Figure 3.5: Schematic presentation of the di�erent scenarios for the RG evolution of the gauge couplingg and their �-functions in the Nf - NC phase space.

If we have that �0 > 0 and �1 > 0, then we have a QED-like theory as shown in Figure 3.5. In these

models the �-function is positive at least up to the perturbation theory can not be performed anymore

(i.e. � = g2=4� & 1). In such a model we can have a Landau pole, where the coupling g can go to in�nity

at a �nite energy scale �L as in QED (see Figure 2.6). We have such a model if the condition is met

�0 > 0) 4

11TR > CA; (3.111)

which is Nf > 11NC=2 in the fundamental representation of SU(NC).

For �0 < 0, �1 > 0 and �2 > 0 with a lower number of �avors Nf than the QED-like theories with

�xed NC , the model can �ow to an interacting conformal �xed point of the renormalization group, i.e. it

is IR-conformal (constant at low energies). If the value of the coupling at that point is less than one such

we can perform perturbation theory (i.e. � = g2=4� � 1), then this �xed point is called a Banks-Zaks

�xed point. At the same time the model is also an asymptotically free theory at high energies as shown

in Figure 3.5. More speci�cally, we determine the �xed point from the �-function of the model in Eq.

(3.108) up to two loops to be

Page 74 of 193


�0g�3

(4�)2+ �1

g�5

(4�)4+O �g�7� = 0)

�� =g�2

4�= �4��0

�1:

(3.112)

If we can arrange �4��0 to be smaller than �1, then we have �� < 1. From this it follows when the

coupling �ows to the IR area where it is conformal, and thus the model is a weakly coupled with the

coupling g�. In the fundamental representation of SU(NC), we have a Banks-Zaks �xed point (�� < 1),

if the number of �avors is between

11

2NC > Nf >

34N3C

13N2C � 3

; (3.113)

where the upper bound comes from requirement that �0 < 0 and the lower bound from the requirement

that �1 > 0.

If we decrease the number of �avors even more such that we have �0 < 0, �1 > 0 and �2 < 0, then

we can obtain walking theories. In these theories, lattice calculations show that there is con�nement

before the coupling reaches the �xed point as shown in left panel in Figure 3.6. In this �xed point we

have that �� = g�2=4� > 1, where the coupling walks (conformal) between the energy scales �TC and

�ETC as in right panel in Figure 3.6. Therefore, the techniquarks are condensed when the coupling walks.

For energies over �ETC the model gets asymptotically free and below �TC the techniquarks and -gluons

con�ned.

g

�

�

g�TC �ETC

Walking

Con�nement

Asymptotic freedom

Figure 3.6: Schematic structure of the �-function (the left panel) and the gauge coupling (the rightpanel) which has a con�nement at low energy, a walking phase between the energy scales �TC and �ETC,and asymptotic freedom at high energies.

For even lower number of �avors (�0 < 0 and �1 < 0), we obtain QCD-like theories where there is

con�nement at low energies and asymptotic freedom at high energies as illustrated in Figure 3.5. In the

fundamental representation of a non-Abelian gauge theory with gauge group SU(NC) we have a QCD-like

theory if the number of �avor is both

Page 75 of 193


Nf <11

2NC and Nf <

34N3C

13N2C � 3

; (3.114)

which come from the inequalities �0 < 0 and �1 < 0, respectively. It is di�cult to show where the

distinction between walking and QCD-like models, where the two theories di�er from one another is,

because the coupling is large. It requires non-perturbative methods to determine this distinction.

However, we can imagine a walking model as shown in Figure 3.6. In such a model we can imagine

that the coupling has walked down to an energy scale which is the same as one of the fermion mass. In

that case, the number of �avors is e�ectively decreased with one. If the theory is still in the walking region

in Figure 3.5, then maybe the coupling will walk again until it reaches the mass of the next fermion.

Thus, the number of �avors is again decreased by one. In this way, the coupling continue until the theory

is moved down to a QCD-like region in Figure 3.5, where the coupling blows up at low energies without

walking.

Let us add an extra term to the Lagrangian in Eq. (3.107), which is the four point operator which

comes from an underlying ETC theory as shown in Figure 3.2 for a heavy gauge boson and in Figure 3.3

for a heavy Higgs boson in previous section about ECT models. Thus, the Lagrangian is now

L = Q �D�Q� 1

4Tr[G��G

�� ]� 1

�2ETC

QLQRqLtR + h.c.; (3.115)

where QL;R = (UL;RDL;R)T are the technifermions and qL = (tL; bL)

T are the third generation of SM

quarks. In such a model we have for decreasing energy below the TC scale, �TC, we get a condensation

of techniquarks, and thus we obtain the top mass term

1

�2ETC

QLQRtLtR !1

�2ETC

hQLQRiTCtLtR � mttLtR; (3.116)

where hQLQRiTC is the techniquark condensate at TC scale, and the top mass renormalized at the TC

energy scale is

mt(�TC) =hQLQRiTC

�2ETC

=4�f3��2ETC

; (3.117)

where f� is the pion decay constant.

The two scales, �TC and �ETC, can be connected using the renormalization group equation (Eq.

(3.108) in [15]) as follows

hQQiETC = exp

� �ETC

�TC

d(ln�) (�(�))

!hQQiTC; (3.118)

where is the anomalous dimension of hQQi (a scaling exponent), which is non-perturbative determined

from the particular technicolor model. If we have a QCD-like asymptotically free gauge theory, then

� 1 at large energies and hQQiETC � hQQiTC.If we have a walking theory as in Figure 3.6, then the coupling is walking from �ETC down to �TC. In

this case, the �ne structure constant � is constant in this conformal window, and therefore the anomalous

dimmension is also constant, i.e. that

Page 76 of 193


exp

� �ETC

�TC

d(ln�)

!= exp

� ln

��ETC�TC

��=

��ETC�TC

� : (3.119)

In this case, the condensate is rescaled as follows

hQLQRiETC =

��ETC�TC

� �2hQLQRiTC; (3.120)

where hQLQRiETC is the condensate at the ETC energy scale �ETC. Thus, the top mass renormalized

at �ETC is

mt(�ETC) =hQLQRiETC

�2ETC

=hQLQRiTC

�2ETC

��ETC�TC

� �2: (3.121)

Thus, the �rst advantage with a walking model is that we can lift the fermion mass by having a large

di�erence between the scales �TC and �ETC, i.e. we wish that the conformal windows are large enough

to generate the di�erent SM fermion masses. We can also make the mass hierarchy between the fermion

generation by having di�erent �ETC for the di�erent fermions.

A problem with the ETC theories is that these theories generate four SM fermions operators, which

can be written as1

�2ETC

qLqRqLqR; (3.122)

that contribute to the �avor-changing neutral currents (FCNCs), and e.g. to the K � �K oscillation

which gives a small violation of CP. The second advantage with a walking theory is that the �ETC can

be adjusted very high without changing the fermion masses according to Eq. (3.121) if the anomalous

dimension is near = 2 otherwise the masses of fermions become too small. Therefore, the FCNCs in

Eq. (3.122) can be suppressed by increasing the di�erent �ETC. These di�erent �ETC give rise to the

parameters in the CKM matrix V qij in Eq. (2.36).

3.5 Weinberg Sum Rules and the S Parameter

The e�ective model described until now has a number of free parameters which are �xed by a associated

underlying dynamics. In this section, we assume that the underlying theory is a four dimensional asymp-

totically free gauge theory with only fermionic �elds transforming according to arbitrary representation

of the gauge group. The Weinberg sum rules (WSR) can be used to reduce the number of unknown

parameters of such a model.

The following discussion below is for the chiral symmetry breaking pattern SU(Nf )L SU(Nf )R !SU(Nf )V, but it can easily be generized to any breaking pattern. To derive these sum rules we de�ne the

time ordered two-point function as the di�erence of vector current and axial-vector current correlation

function

i�a;b�� (q) ��d4xe�iq�x

�hJa�;V (x)Jb�;V (0)i � hJa�;A(x)Jb�;A(0)i� � ihIm�a;b��;V � Im�a;b��;A

i; (3.123)

where a; b = 1; : : : ; N2f � 1 are the �avor labels and the currents are

Page 77 of 193


Ja�;V = qT a �q; Ja�;A = qT a � 5q: (3.124)

where T a are the global SU(Nf ) generators. In the chiral limit (where the masses of the quarks go to

zero), we have that

�a;b�� (q) = (q�q� � g��q2)�ab�(q2); (3.125)

which obeys the unsubtracted dispersion relation

�(Q2) =1

�

� 1

0

dsIm�(s)

s+Q2; (3.126)

where Q2 = �q2 > 0 (Eq. (58) in Ref. [1]). We assume that the underlying theory is asymptotically free

above an energy scale �, therefore the behavior of �(Q2) is the same as in QCD at asymptotically high

momenta, we have that �(Q2) � Q�6 (see Ref. [39]).

Thus, in Eqs. (7.226)-(7.227) in Appendix C-3, by expanding the right-hand side of Eq. (3.126) leads

to the �rst and the second Weinberg sum rule (WSR), which are

1

�

� 1

0

dsIm�(s) = 0;1

�

� 1

0

dssIm�(s) = 0: (3.127)

We break the integration in the WSRs into the region with low lying resonances and the region from

this region up to �. This energy scale � is de�ned such that above this scale asymptotic freedom sets in.

The contribution over � will be negligible.

In the �rst region which extends from zero to a threshold �0, where the integral is saturated by pNGBs,

massive vector and axial vector states. Weinberg assumed in his origin paper in Ref. [40] that there is

only a single narrow resonant state with zero width in the vector and axial-vector spectral functions,

which contribute to the sum rules, i.e.

Im�V (s) = �f2V �(s�m2V ) + : : : ;

Im�A(s) = �f2A�(s�m2A) + �f2��(s) + : : : ;

(3.128)

and totally we have

Im�(s) = �f2V �(s�m2V )� �f2A�(s�m2

A)� �f2��(s) + : : : ; (3.129)

where fV , fA and f� are the vector, axial mesons and the massless pion decay constant, respectively, and

mV and mA are the vector and axial-vector masses, respectively.

By inserting the spectral function with in�nite narrow resonances in Eq. (3.129) into the �rst WSR

in Eq. (3.127), we obtain the relation

f2V � f2A = f2� : (3.130)

A more general relation would replace the left hand side of this relation with a sum over all the vector

and the axial-vector states. This WSR holds for both running and walking dynamics.

In the second region which extends from �0 to � encodes also the conformal properties of the theory,

Page 78 of 193


which is the confornal region. The second WSR receives also important contributions from this conformal

region. According to Eq. (12) in Ref. [41], the second WSR gives the relation

f2Vm2V � f2Am2

A ' a8�2

d(R)f4� ; (3.131)

where a = O(1) which is expected to be a positive coe�cient, and d(R) is the dimension of the represen-

tation of the underlying fermions. As for the �rst WSR, generally the left-hand side of the second WSR

will be a sum over vector and axial states. The two WSRs can be combined, which (see Eq. 7.231 in

Appendix C-3) gives

m2V �m2

A 'f2�f2A

�a

8�2

d(R)f2��m2

V

�: (3.132)

For example in a Nf -�avor model, the EW symmetry is gauged and embedded in the �avor symmetry,

SU(Nf )L SU(Nf )R �= SU(Nf )V SU(Nf )A. When the chiral symmetry breaking happens, then the

�avor symmetry breaks to a pure vectorial symmetry group, i.e. SU(Nf )LSU(Nf )R ! SU(Nf )V. Thus,

the correlation function in Eq. (3.123) is zero after the chiral symmetry breaking. For technicolor models

the EW symmetry will break at the same energy scale as the chiral symmetry breaking. Therefore, the

correlation function is a measure for the EW symmetry breaking. Hence, by knowing the correlation

function we can calculate the Peskin�Takeuchi parameter called S parameter (de�ned in Eq. (7.134) in

Appendix I), which is an EW parameter that describes how much the EW symmetry is broken.

In Eq. (5.10) in Ref. [42] the correlation function is linked to the S parameter (precision parame-

ter). The S parameter is related to the absorptive part of the vector-vector minus axial-axial vacuum

polarization (VV-AA vacuum polarization), which is given by

S = 4

� 1

0

ds

sIm�(s) = 4�

�f2Vm2V

� f2Am2A

�; (3.133)

where Im� is obtained by subtracting the GB contributions from Im�. By using the result in Eq. (3.132)

S ' 4�f2�

�1

m2V

+1

m2A

� a 8�2

d(R)m2Vm

2Af

2�

�: (3.134)

The last term arise from the conformal region (from the scale �0 up to �) is expected to be of the same

order of the two other terms and negative. Thus, it is much reduced relative to QCD-like theories. It is

another advantage having a walking technicolor model, because such a model reduces the S parameter,

which is measured to be S = 0:05�0:10 according to the LEP experiments (from Eq. (10.72) in Ref. [73]).

The S parameters of the various walking technicolor models can be calculated numerically, and thus

it can be tested whether these values are consistent with the experimental data from LEP experiments.

The correlation function in Eq. (3.123) in a strong interacting gauge theory with the currents in Eq.

(3.124) can be calculated by lattice methods from the parameters in the e�ective model which we can

use to calculate the S parameter. The mass and decay constants of the vector and axial-vector particles

can also be calculated numerically, and therefore we have that the S parameter can also be determined

according to Eq. (3.133.

In Ref. [67] the vector and the axial-vector masses are calculated on lattice for the SU(2) gauge theory

Page 79 of 193


with Nf = 2 �avors of fermions in the fundamental representation. The results are mV =fTC� � 13:1(2:2)

and mA=fTC� � 14:5(3:6) (combining statistical and systematic errors), where the pseudoscalar decay

constant is fTC� = vEW = 246 GeV. Thus, the masses are mV � 3:2 TeV and mA � 3:6 TeV. In Ref. [68]

these masses are also been calculated on lattice for the SU(2) gauge theory with Nf = 2 �avors of fermions

in the adjoint representation, i.e. like the MWT model. For these models, the results for the T2-B11

lattice are mV =fTC� � 2:38(31) and mV =mA � 0:67(25), which give the corresponding masses mV � 585

GeV and mA � 874 GeV. These vector and axial-vector particles are lighter than when the fermions are

in the fundamental representation. The masses can still be above the experimental constraints, because

their coupling constants can be corresponding smaller. The decay constants, fV and fA, can not yet be

calculated, because they are harder to calculate than the masses.

In the future, it will be possible to calculate the decay constants and therefore also the S parameter

numerically from the parameters of the e�ective model. In that way we can test the various technicolor

models by comparing these results with the experimental result from the LEP experiments.


We have provided an extension of the Standard Model which embodies minimal walking technicolor mod-

els and their interplay with the particles in the Standard Model, the fermions and the EW gauge bosons.

The extension of the Standard Model consists of the relevant low energy e�ective degrees of freedom,

scalars, pseudoscalars as well as spin-1 particles, which are linked to the underlying minimal walking the-

ory. It is called minimal because we have the minimal number of technifermions gauged under the EW

ground (only two technifermions). The number of technifermions in turn is constrained by electroweak

precision measurements, because a higher number of technifermions contribute correspondingly with a

higher number of loop contributions to the EW parameters in Figure 7.7 in Appendix I and thus larger

EW parameters.

Firstly, we have constructed an underlying model of a technicolor model and its e�ective model with

two technifermions and technigluons both in the adjoint representation of SU(2)TC. Secondly, we have

extended this theory with an extended gauge group SU(NETC) which couples the SM fermions to the

particles in the technicolor model. Thirdly, because of for example the masses of the fermions are too

small compared to experimental results, then we have discussed the possibility of the dynamics of walking

technicolor models. These models can provide the needed larger fermion masses, their mass hierarchy and

the needed suppression of the FCNCs. Finally, the relevant EW parameter called S is been derived which

depends on the parameters of the e�ective model, e.g. the decay constants and masses of the vector and

axial-vector scalars. In future, this parameter can be calculated numerically from the parameters of the

e�ective model, and in that way we can test the various technicolor models. The walking dynamics can

also reduce the S parameter, such that it �t with the experimental result from the LEP experiments.

Page 80 of 193

Chapter 4

Composite Higgs Dynamics

In this chapter, we will provide an uni�ed description of models of composite Higgs dynamics, where the

Higgs can be emerge either as a massive excitation of the condensate in technicolor models or as a pseudo-

Goldstone boson in so-called composite Higgs models. This depends on the way the electroweak symmetry,

GEW = SU(2)LU(1)Y, is embedded in the global symmetry group, G. In previous section and Ref. [1],

we had a technicolor model, where the EW symmetry is broken, SU(2)LU(1)Y ! U(1)Q, simultaneously

with the chiral symmetry breaking, SU(4)! SO(4). The classi�cation of relevant underlying gauge theory

for technicolor models appeared in Ref. [49]. The contrary to these technicolor models are composite

Higgs models, which are classi�ed in Refs. [50, 51], where the unbroken symmetry H must contain the

SM electroweak group GEW.

In the traditional technicolor setup, the Higgs boson is identi�ed with the lightest scalar excitation

of the fermion condensate, e.g. the techni-�. These technicolor models are not able to provide mass

to the SM fermions and therefore a new sector must be added. This new sector can modify the mass

of the technicolor Higgs, typically reducing it as in Ref. [48]. Another possibility is to use vacuum

alignment (discussed in Refs. [16, 17]) to align the vacuum such that the Higgs sector does not break

the EW symmetry. In this case the Higgs boson is identi�ed with one of the Goldstone bosons of the

chiral symmetry breaking. The challenges are not only to provide mass to the SM fermions, but also

to construct a Higgs potential that provides mass to the Higgs Goldstone boson by spontaneously EW

symmetry breaking.

We will mostly follow Ref. [2] in this chapter. We will analyze models consisting of two Dirac fermions

which transform according to the fundamental representation of an SU(2) gauge group. We will investigate

the �avor symmetry breaking pattern SO(6) �= SU(4)! Sp(4) �= SO(5), where the coset SU(4)=Sp(4) �=SO(6)=SO(5) contains �ve Goldstone bosons (GBs). The GBs decompose into (2; 2) + (1; 1) of the

subgroup SO(4) 2 Sp(4). This is because a 5-dimensional irreducible representation of Sp(4) decomposes

into a (2; 2) + (1; 1) of the subgroup SO(4) �= SU(2)1 SU(2)2 according to the decomposition method

with Dynkin diagrams in Appendix J. Therefore, this chiral symmetry breaking pattern allows for a Higgs

doublet. In this analysis, we will investigate the minimal scenario of SU(4)! Sp(4) for both a minimal

81

CHAPTER 4. COMPOSITE HIGGS DYNAMICS

technicolor and for a composite GB Higgs scenario by vacuum alignment.

4.1 The Fundamental Lagrangian

In this model we have the chiral symmetry pattern SU(4) ! Sp(4) with an underlying SU(2) gauge

theory with two Dirac �avors which transform as fundamental representation of the gauge group. The

underlying Lagrangian is

L =� 1

4F a��F

a�� + U(i �D� �m)U +D(i �D� �m)D

=� 1

4F a��F

a�� + iU �D�U + iD �D�D +m

2QT (�i�2)CE�Q+

m

2(QT (�i�2)CE�Q)y;

(4.1)

where F a�� is the �eld strength tensor, U and D are the two fermion Dirac �elds which have the bare

mass m, D� is the covariant derivative, C is the charge conjugation operator working on Dirac indices,

�i�2 is the antisymmetric tensor working on color indices, and the Q vector de�ned in Eq. (3.7). The

antisymmetric vacuum, E�, in Eq. (3.13) is used, which breaks the symmetry of the condensate from

SU(4)! Sp(4).

In the case where the fermion mass is zero, m = 0, the Lagrangian has a global SU(4) symmetry. In

the case m 6= 0, the global SU(4) symmetry is explicitly broken to Sp(4) subgroup. We have that the Q

vector transforms under an in�nitesimal SU(4) transformation as Q ! (1 + i�aT a)Q, where T a are the

15 generators of SU(4) with a = 1; : : : ; 15. Therefore, we have that

m

2QT (�i�2)CE�Q!

m

2QT (1 + i�aT aT )(�i�2)CE�(1 + i�bT b)Q+O(�2) =

m

2QT (�i�2)CE�Q+ i

m

2�aQT (�i�2)C(T aTE� + E�T a)Q+O(�2);

(4.2)

and thus the Lagrangian transforms as

L ! L+ im

2�aQT (�i�2)C(T aTE� + E�T a)Q+ h.c.+ : : : : (4.3)

Thus, the only generators that obey the equations T aTE�+E�T a and leave the Lagrangian in Eq. (4.1)

invariant are precisely the ten Sp(4) generators as shown in Eq. (7.2) and Eq. (7.6) in Appendix A.

Although for m = 0 where the Lagrangian has its full SU(4) symmetry, there will appear a spontaneously

breaking as in QCD, which gives a nonzero vacuum expectation value, hURUL +DRDLi 6= 0. It has the

same structure as the terms containing m in Lagrangian, where the dynamical breaking would also be

SU(4)! Sp(4) as shown in Eq. (3.14). According to the Nambu-Goldstone theorem, we will achieve �ve

GBs from the �ve broken generators.

Page 82 of 193


4.2 Electroweak Vacuum Alignment

We can consider the vacua where the EW sector has been embedded, such that it does not break the EW

symmetry. There are two EW inequivalent valua (discussed in Ref. [26]), which can not be related by an

SU(2)L transformation, which are

�A =

0@ i�2 0

0 i�2

1A and �B =

0@ i�2 0

0 �i�2

1A ; (4.4)

which come from the most general vacuum in Eq. (7.251) which is derived in Appendix C-4. We have

sin(�) = 0 for composite Higgs models and ei� = 1 for � = 0. In this chapter we will use �B.

There is another alignment of the condensate

�H = E� =

0@ 0 1

�1 0

1A ; (4.5)

which breaks the EW symmetry, and thus it can be used to construct technicolor models as in Refs. [52,53],

where we have sin � = 1 and � = 0 in Eq. (7.251).

4.2.1 The �B Vacuum:

We have according to Eq. (3.15) that the unbroken generators of SU(4) for the vacuum �B are de�ned

by

Sa�B +�BSaT = 0; (4.6)

where a = 1; : : : ; 10, because ten of the generators of SU(4) are unbroken. The six of these form an

SU(2) SU(2) subgroup of Sp(4), which are

S1;2;3 =1

2

0@ �i 0

0 0

1A and S4;5;6 =

1

2

0@ 0 0

0 ��Ti

1A ; (4.7)

where we can identify the EW generators with S1;2;3 for SU(2)W and S6 for U(1)Y. Thus, we can

identify the custodial symmetry in SU(2) SU(2) group generated by the unbroken generators S1;:::;6.

The remaining four are

S7;8;9 =1

2p2

0@ 0 i�i

�i�i 0

1A and S10 =

1

2p2

0@ 0 1

1 0

1A : (4.8)

The �ve broken generators which is associated with the �ve GBs are

X1 =1

2p2

0@ 0 �3

�3 0

1A ; X2 =

1

2p2

0@ 0 i1

�i1 0

1A ; X3 =

1

2p2

0@ 0 �1

�1 0

1A ;

X4 =1

2p2

0@ 0 �2

�2 0

1A ; and X5 =

1

2p2

0@ 1 0

0 �1

1A ;

(4.9)

Page 83 of 193


which satisfy the equations (cf. the second relation in Eq. (7.11) in Appendix A),

Xi�B � �BXiT = 0; (4.10)

where i = 1; : : : ; 5. With the above decomposition, we can move in the quotient SU(4)=Sp(4) around the

vacuum �B in following way

� = ei�iXi=f � �B; (4.11)

where the �elds �1;2;3 are the GBs eaten by the massive W� and Z bosons, the �uctuations around the

vacuum of �4 is identi�ed with the Higgs (i.e. h�4i = v) and �5 = � is a singlet scalar.

4.2.2 The �H Vacuum

According to Ref. [52] the unbroken generators of SU(4) for the vacuum �H are

S1 + S4; S2 + S5; S3 + S6; S7;9;10; X1;2;3;5; (4.12)

and the broken ones can be written as

S1 � S4; S2 � S5; S3 � S6; S8 and X4: (4.13)

According to Ref. [52] the vev along the direction �H breaks the SO(4) � SU(2) SU(2) 2 Sp(4) to a

SU(2)C group with the generators S1+S4, S2+S5 and S3+S6, which is in agreement with SM breaking

pattern, where the EW symmetry is broken and we are left with a custodial symmetry. Therefore in this

case, we have a technicolor model, where the whole EW group is not in the unbroken group H = Sp(4).

4.2.3 A Superposition of the two Vacua:

We have now analyze the two vacuum alignment limits, the EW Higgs vacuum alignment limit and the

technicolor limit with the vacua �B and �H, respectively. Several of the results in this subsection can

be found in Ref. [26]. Now, we de�ne the vacuum of the model to be a superposition of these two vacua

above

�0 = cos ��B + sin ��H; (4.14)

where it is normalized in such a way that �y0�0 = 1, and the angle � is a free parameter which is � = 0 for

EW unbroken phase and � = �=2 for a purely technicolor model. According to Eq. (7.20) in Appendix

A, the �ve broken generators can be written in the vacuum �0 as follows

Y 1 = c�X1 � s� S

1 � S4p2

; Y 2 = c�X2 + s�

S2 � S5p2

; Y 3 = c�X3 + s�

S3 � S6p2

; Y 4 = X4 and

Y 5 = c�X5 � s�S8;

(4.15)

where c� = cos � and s� = sin �. These �ve broken generators satisfy the equations

Y i�0 � �0YiT = 0 with i = 1; : : : ; 5: (4.16)

Page 84 of 193


The ten unbroken generators are shown in Eq. (7.21) in Appendix A.

Here are the �rst three generators Y 1;2;3 associated to the GBs that become the longitudinal degrees

of freedom of W� and Z gauge bosons. If we work in unitary gauge we use only the �elds h and �

explicitly which is associated to the generators Y 4;5. Thus, we can write

� = ei(hY4+�Y 5)=f � �0: (4.17)

Here h can be identi�ed as the Higgs boson (for sin � 6= 1). Some studies have � as a composite dark

matter candidate.

The kinetic e�ective Lagrangian of � with interactions to the gauge bosons via minimal coupling is

expanded in Appendix C-4. The kinetic term of � is given in Eq. (7.261) as

f2Tr[(D��)yD��] =

1

2(@�h)

2 +1

2(@��)

2 +1

48f2[�(h@�� @�)2]+

�2g2W+

� W�� + (g2 + g02)Z�Z�

��f2s2� +

s2�f

2p2h

�1� 1

12f2(h2 + �2)

�+

1

8

�c2�h

2 � s2��2��

1� 1

24f2(h2 + �2)

��+O(f�3);

(4.18)

where the covariant derivative of � expressed as follows

D�� = @�� igW a� (S

a�+ �SaT )� ig0B�(S6�+ �S6T ); (4.19)

which is derived in Appendix C-4 in Eqs. (7.263)-(7.266) such that the kinetic-gauge term is invariant

under the gauge transformations. From the expansion above we can identify the masses of the W� and

Z gauge bosons, which are

m2W = 2g2f2s2� and m2

Z = 2(g2 + g02)f2s2� = m2W =cW ; (4.20)

where mW = gv=2 in the SM, thus the vev is

v = 2p2fs�: (4.21)

We can also identify the couplings between the Higgs h and the gauge bosons,

ghWW =gmW c� = gSMhWW c�;

ghZZ =pg2 + g02mZc� = gSMhZZc�;

ghhWW =g2c2�4

= gSMhhWW c2�;

ghhZZ =ghhWW =c2W ;

(4.22)

and couplings between � and the gauge bosons,

g��WW =� 1

4g2s2� = �gSMhhWW s

2�

g��ZZ =g��WW =c2W :

(4.23)

Page 85 of 193


It can be noted that the kinetic term of � is invariant under the Z2 transformation � ! ��, and therefore

� will be stable. Because � is stable, then in Ref. [54] they study � as a composite dark matter candidate.

4.3 Loop Induced Higgs Potential

Generally, we have the �avor symmetry breaking pattern G ! H. In the technicolor limit we always

need that SU(2)W U(1)Y � G and U(1)Q � H. Therefore, the chiral symmetry breaking happens

simultaneously with the EW symmetry breaking, i.e. �TC = vEW. In addition, we need that there can

be found a triplet of GBs which are absorbed as the longitudinal degrees of freedom of the weak gauge

bosons in the quotient G=H. Furthermore, we must identify a custodial symmetry in unbroken symmetry

group SU(2)C � H.

However, in the composite Higgs limit we always need that SU(2)WU(1)Y � G, SU(2)WU(1)Y � H

and again a custodial symmetry in the unbroken symmetry group SU(2)C � H. Therefore, the EW

symmetry is unbroken after the chiral symmetry breaking. Thus, we need a SU(2)W doublet (a Higgs

doublet) in the quotient G=H, which contributes with the SM Higgs boson and the three GBs eaten by

the weak gauge bosons. Therefore, it is needed that we induce a Higgs potential between the energy scale

of the chiral symmetry breaking and the EW symmetry breaking.

In this section we will derive such a Higgs potential, which is induced by gauge one-loops, top-Yukawa

one-loop and an explicit mass term. The dynamics does not tell about where the condensate is aligned in

the SU(4) space in the above theory. As we will see the gauge interaction loop, the top-Yukawa loop and

the loop from an explicit mass term will induce a Higgs potential. The breaking of the �avor symmetry

SU(4) ! Sp(4) will be communicated to the GBs via these loops, which will induce a Higgs potential

that determines the value of the vacuum alignment angle � in Eq. (4.14). These loop-induced potential

for this model has also been calculated in Refs. [26, 55].

4.3.1 Gauge Contributions

We start to derive the contributions to the one-loop potential of the gauge bosons. To do this we construct

the lowest order operator which is invariant under the �avor symmetry SU(4). To construct this operator

we need to write out the kinetic term of � in Eq. (4.18) with its interactions with gauge bosons via

minimal coupling, which yields

f2Trh(D��)

yD��i=

f2Trh(@��)

y@�� igW a�(@��)y(Sa�+ �SaT ) + igW a

� (�ySa + SaT�)@��

ig0B�(@��)y(S6�+ �S6T ) + ig0B�(�yS6 + S6T�y)@��+

g2W a�W

b�(�ySa + SaT�y)(Sb�+ �SbT ) + gg0W a�B

�(�ySa + SaT�y)(S6�+ �S6T )+

g0gB�W a�(�yS6 + S6T�y)(Sa�+ �SaT ) + g02B�B�(�yS6 + S6T�y)(S6�+ �S6T )i;

(4.24)

Page 86 of 193


where the covariant derivative of � is

D�� = @�� igW a� (S

a�+ �SaT )� ig0B�(S6�+ �S6T ): (4.25)

The gauge generators of SU(2)L are S1;2;3, while the one for U(1)Y is S6. The two terms with one W a�

boson cancel each other as follows

Trh� igW a�(@��)

y(Sa�+ �SaT ) + igW a� (�

ySa + SaT�y)@��i

= igW a�Trh� (@��)

y(Sa�+ �SaT ) + (�ySa + SaT�y)@��i

= igW a�Trh� (@��)

ySa�� (@��)y�SaT +�ySa@��+ SaT�y@��

i= igW a�Tr

h� Sa�(@��)y � SaT (@��)y�+ Sa(@��)�y + SaT�y@��

i= igW a�Tr

hSa(@��)�

y � SaT (@��)y�+ Sa(@��)�y � SaT (@��)y�i

= 2igW a�TrhSa(@��)�

y � SaT (@��)y�i

= 2igW a�TrhSa if (@�hY

4 + @��Y5)��y + i

f (@�hY4 + @��Y

5)SaT�y�i= 0;

(4.26)

because ��y = �y� = 1 and therefore

@�(��y) = (@�)�y +�(@��)

y = 0)= �(@��)y = �(@�)�y: (4.27)

The two terms with one B� boson also cancel as follows

Trh� ig0B�(@��)y(S6�+ �S6T ) + ig0B�(�yS6 + S6T�y)@��

i= 2ig0B�Tr

hS6 if (@�hY

4 + @��Y5)��y + i

f (@�hY4 + @��Y

5)S6T�y�i= 0:

(4.28)

Therefore, the kinetic term of � including its interactions with the gauge bosons in Eq. (4.24) can be

written as

f2Trh(D��)

yD��i=

f2Trh(@��)

y@��+ g2W a�W

b�(�ySa + SaT�y)(Sb�+ �SbT )+

gg0W a�B

�(�ySa + SaT�y)(S6�+ �S6T ) + g0gB�W a�(�yS6 + S6T�y)(Sa�+ �SaT )+

g02B�B�(�yS6 + S6T�y)(S6�+ �S6T )i;

(4.29)

which will be used to estimate the gauge loop contribution to the Higgs potential.

� �

q q

p

W 1;2;3� ; B�

�

q

�

q

iM(1)Gauge iM(2)

Gauge

Figure 4.1: In left panel we have the contributions to the one-loop potential of the gauge boson loops,which can be e�ectively drawn as the diagram in right panel.

Page 87 of 193


The contribution to the one-loop potential of the SU(2) gauge boson loops as shown in Figure 4.1 can

be estimated from the following term in the kinetic e�ective Lagrangian, which can be written as

f2g2W a�W

b�Trh(�ySa + SaT�y)(Sb�+ �SbT )

i=f2g2W a

�Wb�Tr

h�ySaSb�+ �ySa�SbT + SaT�ySb�+ SaT�y�SbT

i=f2g2W a

�Wb�Tr

h�ySaSb�� Sa�Sb� � Sa��Sb�+ SaT�y�SbT

i=f2g2W a

�Wb�Tr

hSaSb � (Sb�)�Sa�� (Sa�)�Sb�+ SaSb

i=f2g2W a

�Wb�h�ab � 2Tr

�(Sa�)�Sb�

�i;

(4.30)

where we have used that �T = ��, and SaT = Sa� because the generators are hermitian. The second

term of this expression gives an e�ective vertex with two � �elds and two SU(2) gauge bosons �elds. The

Lagrangian term for this e�ective vertex can be written as

�2f2g2 ~Cg3Xa=1

W a�W

a�Tr [Sa � � � (Sa � �)�] ; (4.31)

where the factor ~Cg is a form factor of the vertex, which only can be determined by non-perturbative

methods, e.g. lattice methods. The external lines of the two gauge bosons can be put together to make

a loop as shown in left panel in Figure 4.1, where there is integrated over the momemtum of the gauge

boson. Thus, we write the one-loop potential of the SU(2) gauge bosons as

VSU(2) =� Cgg2f43Xa=1

Tr [Sa � � � (Sa � �)�] ; (4.32)

which is e�ectively the diagram in right panel in Figure 4.1. The factor Cg is a unknown loop factor.

This one-loop potential is expanded in powers of f up to quadratic terms in the �elds h and � in Eq.

7.273 in Appendix C-4 as follows

VSU(2) =� Cgg2f43Xi=1

Tr�Si � � � (Si � �)��

=Cgg2

��3

2f4c2� +

3

2p2f3c�s�h+

3

16f2(c2�h

2 � s2��2) + : : :

�:

(4.33)

Analogously, the contribution to the one-loop potential of the U(1) gauge boson. The following term in

the kinetic e�ective Lagrangian in Eq. (4.29) can be rewritten to be

f2g02B�B�Trh(�yS6 + S6T�y)(S6�+ �S6T )

i= f2g02B�B�

h1� 2Tr

�(S6�)�S6�

�i: (4.34)

The Lagrangian term for this e�ective vertex can be written as

�2f2g02 ~CgB�B�Tr�S6 � � � (S6 � �)�� : (4.35)

By integrating over the momentum of the loop in left panel in Figure 4.1 gives

VU(1) =� Cgg02f4Tr�S6 � � � (S6 � �)�� : (4.36)

Page 88 of 193


By expanding in powers of f we obtain in Eq. (7.277) in Appendix C-4 that

VU(1) =� Cgg02f4Tr�S6 � � � (S6 � �)��

=Cgg02��1

2f4c2� +

1

2p2f3c�s�h+

1

16f2(c2�h

2 � s2��2) + : : :

� (4.37)

To determining the unknown loop factor Cg in Eq. (4.33) and Eq. (4.37) we can expand the following

trace as follows

Tr�Si � � � (Si � �)�� = cih2 + : : : ; (4.38)

where i = 1; 2; 3 for W a� loops and i = 6 for B� loop, and ci are coe�cients in the front of the quadratic

term of the Higgs h. Thus, from Eq. (4.31) and Eq. (4.32) we have

VSU(2) =� 2f2g2 ~Cg

3Xa=1

W a�W

a�Tr [Sa � � � (Sa � �)�] = �2f2g23Xa=1

W a�W

a�cah2 + : : : ;

VSU(2) =� Cgg2f43Xa=1

Tr [Sa � � � (Sa � �)�] = �Cgg2f43Xa=1

cah2 + : : : :

(4.39)

From the Feynman rules of Eq. (4.39), the amplitudes of the diagrams in Figure 4.1 are

iM(1)Gauge =i2

1

2(�2f2g2 ~Cgcig��)

�d4p

(2�)4(�ig��)p2

= �8f2g2 ~Cgci i

16�2�2 = �i ~Cg f

2g2ci

2�2�2;

iM(2)Gauge =� iCgg2f4ci;

(4.40)

where the integral is solved in Eq. (2.101), and � = 4�f is a cuto� where the condensate is melting.

Because these amplitudes are equal to each other, then we can isolate the unknown loop factor

M(1)SU(2) =M

(2)SU(2) , ~Cg

f2g2ci

2�2�2 = Cgg

2f4ci

)Cg = �2

2�2f2~Cg =

(4�f)2

2�2f2~Cg = 8 ~Cg:

(4.41)

We can �nd the value of � by minimizing the �eld independent term�@Vgauge@�

�h;�=0

=@

@�

�VSU(2) + VU(1)

�h;�=0

= Cg(3g2 + g02)f4c�s� = 0: (4.42)

Because the loop factor Cg is positive, then this part of the potential has a minimum at � = 0. This

minimum does not break the EW symmetry, therefore the vacuum is aligned in the composite Higgs

limit. It can also be noted that the linear term of the Higgs h is always proportional to the derivative of

the potential, and thus this term vanishes at the minimum.

4.3.2 Top Contribution

Now, we will calculate the e�ects on the vacuum alignment from a top-loop contribution to the potential.

We assume that the top mass is generated by the four-fermion operator (cf. Eq. (3.12) in Ref. [26])

yt�2t

(Qtc)y� TP� + h.c.; (4.43)

Page 89 of 193


where � is an SU(2)L index, Q is an SU(2)L doublet, tc is the charge conjugated of the top �eld, are the

technifermions, �t is a new dynamical scale, and the projectors P� select the components of the object

T that transform as an SU(2)L.

t

t

t

t

��TC

Figure 4.2: When the technifermions condense at the energy �TC, then the four-fermion operator in Eq.(4.43) generates a new operator in Eq. (4.45).

These projectors can be written as (cf. Eq. (3.13) in Ref. [26])

P 1 =

0BBBBBB@

0 0 1 0

0 0 0 0

�1 0 0 0

0 0 0 0

1CCCCCCA; P 2 =

0BBBBBB@

0 0 0 0

0 0 1 0

0 �1 0 0

0 0 0 0

1CCCCCCA: (4.44)

When the technifermions condense at �TC, this four-fermion operator generates a new operator as

shown in Figure 4.2 with two top external lines and one � external line. This gives the operator

y0t ~Ctf(Qtc)y�Tr(P

��) + h.c. � �y0t ~Ct�fs� +

1

2p2c�h� 1

16fs�(h

2 + �2) + : : :

�tRt

cL; (4.45)

where y0t is proportional to yt(4�f)2=�2

t , and the factor ~Ct is a form factor of the vertex, which only

can be determined by non-perturbative methods, e.g. lattice methods. The expansion of this operator

generates the top mass, mt = y0tfs�, from the �rst term and the top-Yukawa coupling from the second

term when � 6= 0, which is

�t =y0tc�2p2=mtc�v

; (4.46)

where we have used that v = 2p2fs� from Eq. (4.21).

�

q

�

q

p� qt

t

p

�

q

�

q

iM(1)Top iM(2)

Top

Figure 4.3: In left panel we have the contribution to the one-loop potential of the top quark loop, whichcan be e�ectively drawn as the diagram in right panel.

From the above operator we can construct the contribution of the top-loop to the Higgs potential

by putting two operators together as shown in left panel in Figure 4.3. According to Eq. (7.278) in

Appendix C-4, this contribution is

Page 90 of 193


Vtop =� Cty02t f42X

�=1

[Tr(P��)]2

=� Cty02t�f4s2� +

1p2f3c�s�h+

1

8f2(c2�h

2 � s2��2) + : : :

�;

(4.47)

where there is integrated over the momentum in the top-loop which gives the unknown loop factor Ct.

This loop factor can be determined by expanding the following trace as follows

Tr(P��) = c�h+ : : : ; (4.48)

where c� are coe�cients in the front of the linear term of the Higgs h. Thus, we can rewrite

y0tf(Qtc)y�Tr(P

��) + h.c. = y0tf(Qtc)y�c

�h+ � � �+ h.c.;

Vtop = �Cty02t f42X

�=1

[Tr(P��)]2 = �Cty02t f42X

�=1

[c�h]2 + : : : :(4.49)

From the Feynman rules of Eq. (4.49), the amplitudes of the diagrams in Figure 4.3 are

iM(1)top =(�1)3(iy0tf ~Ctc

�)2�

d4p

(2�)4Tr[i=pi(=p� =q)]p2(p� q)2 = �i3(y0tf ~Ctc

�)241

16�2�2 ~C2

t = �i 3

4�2(y0tfc

�)2�2 ~C2t ;

iM(2)top =� iCty

02t f

4(c�)2;

(4.50)

where the integral is solved in Eq. (2.101), and � = 4�f is a cuto� where the condensate is smelting.

Because these amplitudes are equal to each other, thus we can isolate the unknown loop factor

iM(1)top = iM(2)

top , i3

4�2(y0tf ~Ctc

�)2�2 = iCty02t f

4(c�)2

)Ct = 3�2

4�2f2~C2t = 12 ~C2

t :

(4.51)

As for the gauge loops, we can �nd the value of � by minimizing the �eld independent term of Eq.

(4.47) as follows �@Vtop@�

�h;�=0

= �2Cty02t f4s�c� = 0: (4.52)

The loop factor Ct is positive like the loop factor Cg. Thus, the minimum is located at � = �=2,

which breaks the EW symmetry at TC scale where the technifermions condense. We have the top-loop

contribution to the Higgs potential such that it prefers the vacuum in the direction which corresponds to

the TC vacuum limit.

In the TC vacuum limit (� = �=2), the pNGB h can not be a Higgs-like particle, because the linear

couplings of h to the gauge bosons and to the top vanish according to Eq. (4.22) and Eq. (4.46).

Therefore, the physical Higgs state can not be one of the pNGBs, and it must be the lightest composite

scalar state. The two pNGB h and � can instead be linking together into a complex di-techniquark GB,

which can be written as h+ i�.

We have also in the TC vacuum limit that the masses of h and � are degenerate, and according to

Eq. (4.33), Eq. (4.37) and Eq. (4.47) their loop-induced mass is

Page 91 of 193


m2DM = m2

h = m2� =

f2

4

�Cty

02t � Cg

3g2 + g02

2

�: (4.53)

Thus, the weak gauge interactions misalign the TC vacuum, where the h and � are massive and the three

pNGBs �1;2;3 are massless. While the top-loop corrections realign the vacuum in the TC direction, where

we have the massive h+ i� �eld and the three massless pNGBs �1;2;3. Therefore, the top-loop corrections

provide a positive mass to a potential dark matter candidate h+ i�. The complex state, h+ i�, is a good

dark matter candidate, because it has a U(1) symmetry in the kinetic Lagrangian in Eq. (4.18), and then

it is stable. This state has been used extensively for dark matter model building in Refs. [53, 56�58].

As we have just seen, the pNGB h can not be used as the Higgs boson in the TC-limit. As we

conclude in Eq. (2.168), it gives rise to a problem, because we can not produce a composite particle

with the mass of the Higgs boson, mh = 125 GeV, unless the number of technicolor is very high. This

in turn is constrained by electroweak precision measurements, because a higher number of technicolor

contribute correspondingly with a higher number of loop contributions to the EW parameters in Figure

7.7 in Appendix I and thus larger EW parameters.

Therefore, we want to align the vacuum away from the TC-limit. This can be done by another possible

contribution coming from an explicit term that break the SU(4) �avor symmetry which can give a mass

split between h and �. For example, a mass term of the technifermions that explicitly breaks the SU(4)

�avor symmetry.

4.3.3 Explicit Breaking of SU(4)

Mass terms for the technifermions with a gauge invariant masses can be ad-hoc added to give mass to �.

These are an another sources to the Higgs potential that break explicitly SU(4). Such sources will align

the vacuum away from the TC-limit. A mass term which is aligned with the condensate �B with the

mass M = ��B (cf. Eq. (3.7) in Ref. [26]) can be written as

�� T�B ; (4.54)

which gives the contribution to the Higgs potential, which according to Eq. (7.278) is

Vm =Cmf4Tr(�B � �) = Cm

��4f4c� +

p2f3s�h+

1

4f2c�(h

2 + �2) + : : :

�; (4.55)

where the coe�cient Cm can have both signs. This potential term contributes to push the vacuum away

from the TC-limit (� = �=2). We want to minimize the �eld-independent potential terms of the total

Higgs potential in Eq. (4.33), Eq. (4.37), Eq. (4.47) and Eq. (4.55), which is

V (�) =��3g2 + g02

2Cgc

2� + y02t Cts

2� + 4Cmc�

�f4

=��3g2 + g02

2Cgc

2� + y02t Ct(1� c2�) + 4Cmc�

�f4:

(4.56)

Page 92 of 193


We de�ne that

Xt � y02t Ct �3g2 + g02

2Cg; Xm � Cm: (4.57)

Thus, the �eld-independent potential terms can be rewritten to be

V (�) = Xtc2� � 4Xmc� + constant; (4.58)

which is minimized for

� = 0; c� =2Xm

Xt: (4.59)

We can identify the loop-induced masses of the pNGBs h and � in the Higgs potential in Eq. (4.33),

Eq. (4.37), Eq. (4.47) and Eq. (4.55), which are

m2h =

3g2 + g02

8f2c2�Cg � 1

4f2c2�Cty

02t +

1

2f2c�Cm

=f2

4

�Xt(1� 2c2�) + 2Xmc�

�;

(4.60)

m2� =�

3g2 + g02

8f2s2�Cg +

1

4f2s2�Cty

02t +

1

2f2c�Cm

=f2

4

�Xt(1� c2�) + 2Xmc�

�:

(4.61)

With the solution � = 0, where the EW symmetry is unbroken, the masses read

m2h =

f2

4(2Xm �Xt); m2

� =f2

2Xm: (4.62)

This solution is stable if Xt < 2Xm.

For the solution c� = 2Xm=Xt, the masses read

m2h =

f2

4

X2t � 4X2

m

Xt=f2

4s2�Xt; m2

� =f2

4Xt: (4.63)

For m2h > 0 we need that Xt > 2jXmj (i.e. c� < 1) which corresponds to broken EW symmetry. We

recover the relation m2� = m2

h=s2� as in Ref. [26].

The pNGB Higgs mass can be rewritten to

m2h =

f2

4s2�Xt =

f2

4s2�

�y02t Ct �

3g2 + g02

2Cg

�=f2

4s2�

�m2t

f2s2�Ct � 2m2

W +m2Z

4f2s2�Cg

�

=Ctm

2t

4

�1� 2m2

W +m2Z

4m2t

CgCt

�=Ctm

2t

4

�1� 2(80:385 GeV)2 + (91:188 GeV)2

4(172:44 GeV)2CgCt

�

=Ctm

2t

4

�1� 0:179

CgCt

�;

(4.64)

where we have used that the top mass mt = y0tfs� and Eq. (4.20). This shows that the contribution from

the gauge loops is typically smaller than the top-loop, if we are assuming that Cg � Ct. If we neglect the

contribution from the gauge loops, then we have

m2h �

Ctm2t

4) Ct � 4m2

h

m2t

=4(125:09 GeV)2

(172:44 GeV)2� 2: (4.65)

Therefore, we have that the loop factor in the top-loop contribution is Ct � 2. The Higgs couplings to

Page 93 of 193


the gauge bosons in Eq. (4.22) are now well constrained by LHC data (see Eq. (2.44)). Thus, a realistic

value of � must be small, i.e.

c� � 1) Xt � 2Xm: (4.66)

Thus, the vacuum is aligned close to the composite Higgs limit (i.e. � = 0). In this case, we can produce

small enough mass of the Higgs boson, and large enough mass gap up to the next lightest resonance (the

pNGB �) with the mass m2� = m2

h=s� (cf. Eq. (4.63)) because s� is small. With this vacuum alignment

there is no more a dark matter candidate, because the pNGB � is not stable anymore. This is, because

the added potential has no Z2 symmetry of � for higher expansion order of the potential, and therefore �

can decay. However, another possible problem arises, because the vacuum alignment angle s� is needed

to be �ne-tuned.

4.4 Fine-Tuning of the Model

The de�nition in Eq. (2.116) of the quantity for how much a observable O is �ne-tuned compared to a

parameter �i is

�OBG;i �

��iO @O@�i

�� < �max; (4.67)

where for example we can choose the maximal tolerance for �ne-tuning to be �max = 100. We can

calculate this quantity for the �ne-tuning of the top-coupling, y0t, to the vacuum alignment angle, s�, in

the above model. We isolate s2� in Eq. (4.59), which gives

c� =2Xm

Xt, s2� = 1� 4X2

m

X2t

: (4.68)

This gives the �ne-tuning quantity

�s2�BG;y02t

��y02ts2�

@s2�@�2t

�� =��y02ts2�

@Xt

@y02t

8X2m

X3t

�� = y02ts2�Ct

8X2m

X3t

: (4.69)

By inserting Ct � 2 and Xt � 2Xm = 2Cm (cf. Eq. (4.65) and Eq. (4.66) respectively) into this

�ne-tuning quantity, we obtain

�s2�BG;y02t

� 2y02tCm

1

s2�; (4.70)

where the coe�cient in the front of 1=s2� is roughly at the order of unity. Thus, the �ne-tuning quantity

is very large, because s� is very small cf. Eq. (4.66). Therefore, we need to �ne-tune the parameter s�

very much. According to Eq. (4.66) we need to �ne-tune between the top-loop and the explicit breaking

of SU(4) contributions. This �ne-tuning must be induced by another completely di�erent mechanism.


In this chapter, we have constructed a model of a composite Higgs based on a strongly interacting gauge

theory with fermionic matter �elds, where we have studied simultaneously models of pNGB Higgses and

Page 94 of 193


TC models. In the TC limit the Higgs is identi�ed with the lightest scalar resonance, techni-�, of the

dynamics, while away from the TC limit it is identi�ed with one of the pNGBs, h.

We have focused on the example of the �avor symmetry pattern, SU(4) ! Sp(4). We have stud-

ied the most minimal strongly coupled gauge theory, the SU(2) gauge theory, with two technifermions

transforming as a fundamental representation of the gauge group. The coset SU(4)=Sp(4) contains �ve

pNGBs, where three of them are eaten by the massive weak gauge bosons. The fate of the remaining

pNGBs depends on the alignment of the vacuum. In the TC alignment, they form a complex dark matter

candidate, h+ i�. In the alignment away from the TC-limit one of the two pNGBs plays the role as Higgs

boson, while the another pNGB can not play the role as dark matter state, because it is not expected to

be stable.

Our analysis shows that the alignment in the TC-limit is more natural, because there is small or

no �ne-tuning between the contribution from the top-loop and the explicit breaking term and thus the

vacuum alignment angle s�. However, in TC-limit there is the problem that the Higgs boson must be the

lightest scalar resonance, techni-�, which seems to have too much mass to be identi�ed with the Higgs

boson. What we have won with this kind of models are that we have got a dynamical explanation of the

scalar nature of the Higgs boson and simultaneously we have removed the problem with a too heavy Higgs

candidate as in the TC models. We obtain these advantages at expense of a new �ne-tuning problem

of the vacuum alignment and that we need to add a Higgs potential ad hoc. Additionally, we have still

not completely removed the EW hierarchy problem, because we need one or more scalars to produce the

fermion masses as in the TC models.

Page 95 of 193

Chapter 5

Partially Composite Higgs Dynamics

We will in this chapter examine the EW symmetry breaking based on the mixture of a fundamental Higgs

doublet, H, and an composite pseudo-Nambu Goldstone doublet. The condensation of the strongly in-

teracting fermions triggers a vev for the fundamental Higgs doublet so that the EW symmetry breaking

still arises dynamically but the EW scale and the Higgs particle arises as a mixture of composite and fun-

damental sectors. This idea is due to 't Hooft in Ref. [11] and were originally termed bosonic Technicolor

(BTC) models (mentioned later in Ref. [71]). Bosonic because of the fundamental Higgs boson doublet

and technicolor because of the composite doublet. However, just as above where we studied CH models

as a misalignment of a TC model, also here we will study such a misalignment with a fundamental scalar

present, as is done in Ref. [3] and term in partically composite Higgs (PCH). In this chapter the possible

triviality and the vacuum stability of this model will be investigated by calculating the running of the

fundamental Higgs self-coupling, �h. From the running we can determine the energy scale, where we have

a Landau pole or an unstable vacuum. In that way we can investigate in what part of the parameter

space the model is self-consistent.

5.1 The Fundamental Lagrangian

In the Ref. [3], the authors consider a minimal non-supersymmetric BTC model with a single fundamental

Higgs doublet,H, and realign the vacuum into a PCHmodel via an electroweak preserving mass term. The

minimal TC sector contains technifermions transforming as fundamental representations under the gauge

group SU(2)TC, where the left- and right-handed technifermions transform as doublets and singlets under

SU(2)W, respectively. Their gauge quantum numbers are shown in Table 5.1. Thus, the technifermions

transform in the fundamental representation of the technicolor gauge group SU(2)TC.

When the weak interaction is turned o�, then the model has a global SU(4) under which the four-

component object

QL = (ULDL~UL ~DL)

T (5.1)

transforms in the fundamental representations explained before where we construct the four-component

96

CHAPTER 5. PARTIALLY COMPOSITE HIGGS DYNAMICS

vector in Eq. (3.7). The condensate of the technifermions can be written as

hQaLQT;bL �C�1i / �ab0 ; (5.2)

where � acts on the TC indices, C�1 = diag(i�2; i�2; i�2; i�2) acts on the left-handed Weyl spinors in QL,

and the most general Sp(4) vacuum is given by

�0 =

0@ ei�� cos � 12 sin �

�12 sin � �e�i�� cos �

1A ; (5.3)

which is derived in Appendix C-4. The angle � 2 [0; �], � is a phase which violates CP, and �0 breaks

SU(4) spontaneously to its subgroup Sp(4), as explained in Appendix C-4. For sin � = 1 the condensate is

purely SU(2)L breaking (the technicolor limit), while for sin � = 0 the electroweak symmetry is unbroken

(the composite-Goldstone limit).

SU(2)TC SU(2)W U(1)Y

(UL; DL)T 2 2 0

~UL 2 1 �1=2~DL 2 1 +1=2

Table 5.1: Technifermion gauge quantum numbers.

The kinetic terms for the SM fermions , technifermions QL including the electroweak and TC

interactions, and all gauge �elds are written as follows

LK =i � �D� + iQyL��(14@� � iA� � iGa��a=214)QL �

1

4F a��F

a�� (5.4)

where �� = (1;�~��) and the covariant derivative is given by

D� = @� � ig1Y2B� � ig2 �

a

2W a� � ig3

�a

2Ga�: (5.5)

The elektroweak (EW) gauge bosons are included in the matrix

A� =

0@ g2W

a�12�

a 0

0 �g1B� 12�

3

1A ; (5.6)

and F a�� = @�Aa� � @�Aa�+ gF abcAb�A

c� which is the �eld strength tensor of one of the gauge �elds, where

F abc = 0 for the photon �eld, F abc = �abc for the W�, Z and the technigluon �elds, and F abc = fabc for

the gluon �elds.

The new in this PCH model compared to the TC and CH models in the previous two chapters is a

Higgs sector with a fundamental Higgs doublet, H. The kinetic Lagrangian for this Higgs doublet and

its potential in SU(4) notation is given by

Page 97 of 193


LH =1

2Tr[(D�H)yD�H]� V (H)

=1

2Tr[(D�H)yD�H] +QTL�C

�1(M + �)QL + h.c.�m2H jHj2 � �hjHj4:

(5.7)

The fundamental Higgs doublet H has a positive mass parameter m2H and the quartic self-coupling, �h,

and can be written (same form as the Higgs doublet in the SM in Eq. (2.18)) as

H(x) =1p2

0@ i�1h(x) + �2h(x)

v + �h(x)� i�3(x)

1A ; (5.8)

where the �elds �h (~�h) are the scalar components of H with the vacuum expectation value (vev) v �jhHij. The gauge covariant derivative reads

D� = @� � ig1Y2B� � ig2 �

a

2W a� : (5.9)

The matrices M and � are two 4 � 4 matrices, which contain the gauge-singlet masses m1;2 and the

Higgs-Yukawa couplings to the two technifermions �U;D, and can be written as follows

M =1

2

0@ m1� 0

0 �m2�

1A ; � =

1

2

0@ 0 �H�

HT� 0

1A ; (5.10)

where

H� =1p2

0@ �U (�h + v� � i�3h) �D(�i�1h + �2h)

��U (i�1h + �2h) �D(�h + v + i�3h)

1A : (5.11)

The Yukawa terms with the technifermions come from the second term with the technifermions in Eq.

(5.7). These terms give rise to the technifermion masses, which are mU = �Uv�=p2 and mD = �Dv=

p2.

The mass terms can be written as

m1ULDL +m2~UL ~DL +mUUL ~UL +mDDL

~DL; (5.12)

where the terms with the masses m1 and m2 are terms which can be written because they are gauge

invariant. To explain these terms we need a new scalar, call it S. This must be an EW singlet, and hence

we can not use the scalar H. Therefore, we need to construct terms like �1SULDL and �2S ~UL ~DL, where

the masses are m1 = �1vS=p2 and m2 = �2vS=

p2.

The last part of the total Lagrangian is the Yukawa Lagrangian terms of the fundamental Higgs H

to the SM fermions. The Yukawa Lagrangian can be written according to Eq. (2.73) in terms of Weyl

spinors as follows

LY =� �u"ijqLjHiu�R � �dq�LiHidR � �elLiHie

�R + h.c.; (5.13)

where �u, �d and �e are the Yukawa couplings, and qL = (uL; dL)T and lL = (�L; eL)

T are the left-handed

weak doublets of the quarks and the leptons, respectively, and uR, dR and eR are the right-handed weak

singlets of the up-type quarks, the down-type quarks and the electron-type leptons respectively.

Finally, we can collect all Lagrangian terms above in the total fundamental Lagrangian of this theory,

Page 98 of 193


which is

L =i � �D� + iQyL��(@� � iA� � iGa��a=214)QL �

1

4F a��F

a��

+1

2Tr[(D�H)yD�H] +

�QTL�C

�1(M + �)QL + h.c.��m2

H jHj2 � �hjHj4��u"ijqLjHiu

�R � �dqyLiHidR � �elLiHie

�R + h.c.

�:

(5.14)

5.2 Construction of the E�ective Lagrangian

Initially, we have the global symmetry group G = SU(4), which is spontaneously broken to the global

group H = Sp(4). The SU(4)=Sp(4) coset contains �ve broken generators, Xa, and ten unbroken gen-

erators, Si, which satisfy the relations Xa�0 � �0XaT = 0 and Si�0 + �0S

iT = 0 cf. Eq. (7.11) in

Appendix A. The broken and the unbroken generators, Xa and Si, are listed as T a? and T ik in Eq. (7.20)

and Eq. (7.21) in Appendix A, respectively.

The 5-dimensional irreducible representation of Sp(4) can be decomposed into a (2; 2) + (1; 1) under

the subgroup SO(4) �= SU(2)1 SU(2)2 of Sp(4) �= SO(5) as shown in Appendix J by using Dynkin

diagrams. The SU(2)1;2 can be identi�ed with the generators (Sa � Sa+3)=p2 with a = 1; 2; 3. These

generators are reduced to SU(2)L;R generators in Eq. (3.34) in the sin � ! 0 limit. Therefore, Sa with

a = 1; 2; 3 form the isospin group SU(2)V = SU(2)1+2 = SU(2)L+R. The exponential realization of this

5-dimensional representation of Sp(4) �= SO(5) of the Nambu Goldstone bosons �a can be written as

� = exp(p2i�aXa=f); (5.15)

where parameter f is the TC decay constant in the chiral limit. The object � parametrizes in the coset

G=H, while the exponent of � parametrizes in the algebra. This object transforms as � ! U�V y(U; �)

according to Eq. (7.159) in Appendix K, where the global transformations are U 2 SU(4) and V 2Sp(4). We use the realization � which transforms as � ! U�V y instead of the non-linear representation

� = ��0�T (the same � as in Eq. (4.17) in Chapter 4) which transforms as � ! U�UT , because �

transforms both with U and V , and therefore the symmetries are more explicit, and we wish to know how

the objects transform under V 2 Sp(4). Therefore, these building blocks can be used as blocks without

thinking about that further.

We have a freedom to choose between these two representations, � = exp(p2i�aXa=f) and � =

��0�T , because of the powerful �eld theoretic theorem called Haag's theorem (page 101 in Ref. [14]). It

states that there is a representation independence, if two �elds are related non-linearly, e.g. ' = �F (�)

with F (0) = 1, as the above representations, then the same experimental observables result. Here we

have small �uctuations of the �elds ' and �, i.e. we can expand

F (�) = F (0) + �@F

@�

��=0

+ : : : : (5.16)

If this is the case, you can write all e�ective Lagrangian terms in terms of the speci�c representation

which respect the same global symmetries and gauge symmetries, and then you will obtain the same

Page 99 of 193


physics from these terms. E.g. for scalar �elds, the mass terms in terms of the �eld ' can be rewritten

in terms of the other representation � to �rst order in F (�) as follows

m2'2 = m2�2F (�)2 = m2�2(F (0) + : : : ); (5.17)

and also for the kinetic terms

1

2(@�')

2 =1

2((@��)F (�) + �@�F (�))

2=

1

2(@��)

2 + : : : : (5.18)

This is also shown for the interaction terms in Ref. [69].

By using this representation, the kinetic Lagrangian terms are expressed in terms of the quantity

called the Maurer Cartan 1-form (Eq. (7.161) in Appendix K)

C� = i�yD��; (5.19)

which lives in the algebra. The semi-covariant derivative is

D�� = @�� iA��; (5.20)

with the gauge �elds A� in Eq. (5.6). The quantity C� transforms like a Sp(4) gauge �eld as C� !V (C� + i@�)V

y (see in Appendix K). We can project C� onto �elds parallel and perpendicular to the

unbroken Sp(4) direction as (cf. Eq. (7.162) in Appendix K)

C?� = 2Tr(C�Xa)Xa

Ck� = 2Tr(C�Si)Si;

(5.21)

which are a 5-plet and 10-plet of Sp(4), respectively, and C� = C?� +Ck�. These transform homogeneously

and like a gauge �elds according to Eq. (7.165) and Eq. (7.166), i.e.

C?� ! V C?� Vy;

Ck� ! V (Ck� + i@�)Vy:

(5.22)

Furthermore, we can de�ne the quantity

�� = �T (M + �)��0 � h.c.; (5.23)

which transforms under Sp(4) as �� ! V ��V T (Eq. (7.281) in Appendix C-5).

With these building blocks in hand we can construct a Sp(4) invariant Lagrangian. The leading O(p2)chiral Lagrangian is

L(2) = f2

2Tr(C?� C

?�) + 4�f3Z2Tr(�+); (5.24)

where Z2 � 1:47 according to a Nc = Nf = 2 lattice study in Ref. [31]. The �rst term in the TC e�ective

Lagrangian in Eq. (5.24) and the Higgs kinetic term in Eq. (5.7) yield the EW scale (vEW = 246 GeV)

Page 100 of 193


v2EW = f2 sin2 � + v2; (5.25)

which is derived in the Eqs. (7.282)-(7.289) in Appendix C-5. The angle � is the same angle as in the

vacuum matrix in Eq. (5.3), which tells about which direction the vacuum is aligned.

5.3 The Vacuum Alignment

We will minimize the O(p2) potential

V(2)e� = 8�f3Z2

hm12 cos � � �UDv sin �=

p2i+m2H

2v2 +

�h4v4; (5.26)

which is derived in Eqs. (7.290)-(7.301) in Appendix C-5 from second TC Lagrangian term in Eq. (5.24)

and the fundamental Higgs potential in Eq. (5.7). We have de�ned that m12 � m1+m2, �UD � �U +�D

and mUD � mU +mD = v(�U + �D)=p2 � v�UD=

p2. The minimizing of the potential satis�es

@V(2)e�

@�=8�f3Z2

h�m12 sin � � �UDv cos �=

p2i= 0; (5.27)

@V(2)e�

@v=� 8�f3Z2�UD sin �=

p2 +m2

Hv + �hv3 = 0: (5.28)

From the �rst vacuum condition we obtain

tan � = �mUD

m12: (5.29)

The mU;D mass terms tend to align the vacuum in the direction of the TC vacuum limit (� = �=2) as

the top-loop potential in Eq. (4.47). On the other hand, the m1;2 mass terms prefer the direction of the

EW-unbroken vacuum limit (� = 0). They correspond to EW preserving mass operators, as opposed to

the Dirac mass terms mU;D as seen in Eq. (5.12), similarly to the explicit mass term that break the SU(4)

symmetry in Eq. (4.55). From the second vacuum condition in Eq. (5.27), we obtain an expression for

the Higgs self-coupling

�h =4p2�Z2f

3 sin � �m2Hv

v3: (5.30)

In Table 5.3 the important expressions above and their origins are collected.

Expression The Origin of the Expression

tan � = �mUD

m12Vacuum alignment

�4p2�f3Z2�UD sin � +m2Hv + �hv

3 = 0 Vacuum alignment

tan� � vf sin � De�nition

v2EW = f2 sin2 � + v2 The TC and Higgs gauge-kinetic terms

Table 5.2: Important expressions and their origins in this partially composite Higgs model.

Page 101 of 193


5.4 Scalar Resonances

In this section, we determine the masses of the various scalar resonances. In Eqs. (7.302)-(7.319) in

Appendix C-5 the mass matrices of the scalars are calculated explicitly from the TC e�ective Lagrangian

term 4�f3Z2Tr(�+) and from the terms of the fundamental Higgs potential �m2H jHj2��hjHj4. In Eqs.

(7.320)-(7.334) the mass eigenstates and their masses are derived by diagonalizing the mass matrices

below.

According to Eq. (7.313), the charged scalar mass matrix in the basis (�+h ; �+) is

M2�+ =

0@ m2

H + �hv2 �m2

Ht� � �hv2t��m2

Ht� � �hv2t� m2Ht

2� + �hv

2t2�

1A = (m2

H + �hv2)

0@ 1 �t��t� t2�

1A ; (5.31)

where ��h = (�1h � i�2h)=p2 and �� = (�1 � i�2)=p2 and

tan� � t� � v

f sin �: (5.32)

The mass eigenstates of the charged scalar mass matrix in Eq. (5.31) are the two charged pion states

G� = s��h + c��

� and ~�� = �c��h + s��; (5.33)

with the masses (Eq. 7.330)

m2G� = 0 and m2

~�� = (m2H + �hv

2)=c2� : (5.34)

According to Eq. (7.318), the neutral scalar mass matrix in the basis (�3h; �3) is

M2�3 =

0@ m2

H + �hv2 �m2



2� + �hv

2t2�

1A = (m2

H + �hv2)

0@ 1 �t��t� t2�

1A : (5.35)

The mass eigenstates of the other neutral scalar mass matrix in Eq. (5.35) have the same form as the

two charges pion states, which are

G3 = s��3h + c��

3 and ~�3 = �c��3h + s��3 (5.36)

with the masses

m2G3 = 0 and m2

~�3 = (m2H + �hv

2)=c2� : (5.37)

The mass of the �5 which does not mix with the other scalars is according to Eq. (7.319)

m2�5 = t2�(m

2H + �hv

2); (5.38)

Finally, according to Eq. (7.309) we have that the neutral scalar mass matrix in the basis (�h; �4) is

given by

M2h = m2

H

0@ 1 �c�t��c�t� t2�

1A+ �hv

2

0@ 3 �c�t��c�t� t2�

1A ; (5.39)

Page 102 of 193


The mass eigenstates of the neutral Higgs mass matrix in Eq. (5.39) are

h1 = c��h � s��4 and h2 = s��h + c��4: (5.40)

According to Eq. (7.325) in Appendix C-5, the mixing angle � between the two components of h1;2 is

t2� =2c�t�(1 + "=3)

1 + "� t2�(1 + "=3): (5.41)

where " � 3�hv2=m2

H . For " = 0 (�h = 0) we obtain

tan 2� = cos �2 tan�

1� tan2 �= cos � tan 2�; (5.42)

which is the expression below Eq. (24) in Ref. [3]. The masses of h1;2 (cf. Eq. (7.322)) are

m2h1;h2 =

m2H

2

�1=c2� + "(1 + t2�=3)�r�

1=c2� + "(1 + t2�=3)�2� 4

��c2�t2�(1 + "=3)2 + t2�(1 + ")(1 + "=3)

��:

(5.43)

The h1 and h2 mass states are the light and heavy neural Higgs, respectively. The h1 is the candidate

to the Higgs in the SM, which is a linear combination of the fundamental Higgs �h and the composite

pNGB component �4. For small " (3�hv2 � m2

H) and small s� we obtain from Eq. (5.43) (see in Eq.

(7.327)

m2h1 = m2

H

�s2�s

2� + "

2

3s2�(2� c2�)

�+O("2): (5.44)

For �h = 0 (" = 0) we obtain that

m2h1 = m2

H sin2 � sin2 �; (5.45)

in accordance with Eq. (26) in Ref. [3] in the limit s2�c2� � 1. The mass of the Higgs state h1 depends

on sin �, and therefore it acquires its mass from a strong sector vacuum misalignment like in composite

pNGB Higgs models. From Eq. (5.43) we can isolate the self-coupling, which gives

�h =a(b+ c); (5.46)

where the coe�cients are

a =1=(t2�v4(2c2� � 6));

b =� v2(3m2h1 � 4m2

Ht2� + 2c2�m

2Ht

2� +m2

h1t2�);

c =� v2qm4h1(9� 6t2� + t4�) +m2

Hm2h1(8c2�t

2� � 12t2�) + 4(c2�m

4h1t2� +m4

Ht4� +m2

Hm2h1t4�);

(5.47)

which is renormalized at the SM Higgs mass mh1 , i.e. �h = �h(mh1).

The mass eigenstates of the scalars above and their masses in Eqs. (5.38)-(5.37) are collected in Table

5.3. The mass eigenstates G� and G3 are the NGBs that become absorbed as the longitudinal degrees

of freedom of the weak gauge bosons.

Page 103 of 193


The Mass Eigenstate The Mass

h1;2 = c��h � s��4 Equation (5.43)

~�� = �c��h + s�� m2

~�� = (m2H + �hv

2)=c2�

G� = s��h + c��

� m2G� = 0

~�3 = �c��3h + s��3 m2

~�3 = (m2H + �hv

2)=c2�

G3 = s��3h + c��

3 m2G3 = 0

�5 m2�5 = t2�(m

2H + �hv

2)

Table 5.3: The mass eigenstates and their masses in this partially composite Higgs model.

5.5 The Normalization Factors

The coupling of the light Higgs (h1) to the weak gauge bosons (V = W�; Z) h1V V and the Yukawa

coupling to the light Higgs (h1) h1 �ff are normalized to the SM ones as follows�gPCH�V V �h +

1

2c�s�fg

22�

4

�W+� W

�� =�V gSMhV V h1W

+� W

��; (5.48)

�PCHf �hff =�F�SMf h1ff; (5.49)

where gPCH�V V = g22v=2, gSM�V V = g22vEW=2, �

PCHf = mf

p2=v and �SMf = mf

p2=vEW. First and second

term in Eq. (5.48) come from the gauge-kinetic Lagrangian for the fundamental Higgs in Eq. (5.7) and

the �rst term in Eq. (5.24), respectively. We can see that both �h and �4 couple to the weak gauge

bosons, while it is only �h that couples to the fermions. The normalization factors are (Eq. (7.348))

�V = c�s� � s�c�c�;�F = c�=s� ;

(5.50)

which are derived in Eqs. (7.335)-(7.348) in Appendix C-5.

5.6 The Angles in the Model

The signs of sine, cosine and tangent of both the angle �, � and � and the reasons to these signs are

shown in Table 5.4

For example, sin� > 0 and cos� > 0, because we assume that the top-Yukawa couplings are positive.

Firstly, we can assume that sin� � 1, if the top-Yukawa coupling in the model is nearly the same as in

the SM according to Eq. (7.343) in Appendix C-5. Secondly, we can also assume that sin�� 1, because

the composite part �4 in the mass eigenstate of the SM Higgs h1 in Eq. (5.40) is most dominating. Thus,

tan� > 0. The rest of the signs are explained in the table above.

Page 104 of 193


Sign of Angle Reason

sin� > 0 Positive Yukawa Couplings: ��ht = �SMt =s� > 0

tan� > 0 tan� = v=(f sin �) = s�=q1� s2� > 0

cos� > 0 sin �; tan� > 0

sin � > 0 sin � = (f=v) tan� > 0

tan � < 0 tan � = �mUD=m12 < 0

cos � < 0 sin � > 0, tan � < 0

cos� > 0 Positive Yukawa Couplings: �PCHt = (cos�= sin�)�SMt > 0

tan� > 0 sin� � 1) tan 2� < 0) tan 2� = cos � tan 2� > 0 and sin�� 1

sin� > 0 cos� > 0, tan� > 0

Table 5.4: The signs of sine, cosine and tangent of the angles �, � and � and the reasons.

5.7 The Parameter Space

Now, we will investigate the parameter space of this model. This is done by using Matlab, which can

calculate the parameters above from the three input parameters: the mass of the fundamental Higgs mH ,

the angle s� and the angle t� .

Some examples of vacuum alignment are shown in Table 5.5, where the values of the di�erent parame-

ters are calculated from the input parameters mH =pm2H , s� and t�. In these calculations it is used that

mt = 172:44 GeV, mh1 = 125:09 GeV and �s(mZ) = 0:1184 from Table 2.3. Therefore, g3(mt) = 1:1715

and �t(mt) = 0:9319 according to the right panel of Figure 2.6 and Eq. (2.90), respectively. In Table 5.5,

we consider two di�erent masses of the fundamental Higgs, mH = 1000 GeV and mH = 300 GeV. For

each of these masses we have three di�erent values of the angle s� (0.30, 0.15 and 0.05 for mH = 1000

GeV and 0.45, 0.30, 0.15 for mH = 300 GeV) and three di�erent values of the angle t� (3.18, 1.07 and

0.71 for both masses mH and each value of s�). Our choice of the values of t� are random and have no

special meaning.

The fundamental Higgs self-coupling at the mass of the SM Higgs mass �h(mh1) decreases, when

the angle t� increases for �xed values of mH and s�. Thus, there is an upper bound of t� , where the

self-coupling �h(mh1) becomes negative, and therefore the vacuum is unstable. This upper bound of t�

increases with decreasing s�. For example, for the �xed values, mH = 1000 GeV and t� = 3:18, the Higgs

self-coupling �h(mh1) is negative for s� = 0:15 and positive for s� = 0:05. In addition, this upper bound

of t� also increases with decreasing mH . For example, for the �xed values, s� = 0:15 and t� = 3:18, the

Higgs self-coupling �h(mh1) is negative for mH = 1000 GeV and positive for mH = 300 GeV.

Page 105 of 193


mH s� t� s� f m12 �UD v m~� mh2 m�5 �h(mh1) log10�E0

GeV

�1000 0:30 3:18 0:29 246 1024 1:941 235 3275 3262 3124 �0:634 �1000 0:30 1:07 0:67 561 50:8 0:126 180 1440 1414 1051 �0:916 �1000 0:30 0:71 �0:59 667 19:3 0:060 143 1216 1191 707 �1:066 �1000 0:15 3:18 0:30 492 549 0:502 235 3329 3326 3176 �0:048 �1000 0:15 1:07 0:68 1122 27:2 0:033 180 1465 1461 1068 0:1041 2:747

1000 0:15 0:71 �0:57 1334 10:4 0:016 143 1236 1240 719 0:5816 �1000 0:05 3:18 0:30 1476 186:5 0:056 235 3346 3345 3191 0:1332 6:157

1000 0:05 1:07 0:69 3365 9:3 0:004 180 1472 1476 1074 0:4195 6:738

1000 0:05 0:71 �0:57 4003 3:5 0:002 143 1243 1254 722 1:091 �

300 0:45 3:18 0:27 164 133:2 0:405 235 997 988 951 �0:0109 �300 0:45 1:07 0:70 374 7:0 0:028 180 450 443 328 0:1446 2:990

300 0:45 0:71 �0:49 445 2:9 0:014 143 397 414 231 0:7110 �300 0:30 3:18 0:29 246 100:1 0:190 235 1024 1020 976 0:0780 3:712

300 0:30 1:07 �0:69 561 5:3 0:013 180 464 470 338 0:3295 4:714

300 0:30 0:71 �0:48 667 2:2 0:007 143 411 444 239 1:075 �300 0:15 3:18 0:30 492 53:6 0:049 235 1041 1040 993 0:1361 6:368

300 0:15 1:07 �0:68 1122 2:8 0:003 180 472 486 345 0:4469 7:970

300 0:15 0:71 �0:47 1334 1:2 0:002 143 420 463 244 1:3120 �

Table 5.5: Examples of vacuum alignment, scalar spectrum, and the vacuum instability energies E0 (allmasses are in GeV). The parameters are calculated by Matlab. In these calculations it is used thatmt = 172:44 GeV, mh1 = 125:09 GeV and �s(mZ) = 0:1184 from Table 2.3. Therefore, g3(mt) = 1:1715and �t(mt) = 0:9319 according to the right panel of Figure 2.6 and Eq. (2.90), respectively.

The masses of the pNGBs are experimentally constrained, for example the mass of �5 which is a pure

composite particle. The mass of �5 decreases both with decreasing mH and s�, respectively. For example

for �xed, s� = 0:30 and t� = 0:71, the mass of �5 is 707 GeV for mH = 1000 GeV and 239 GeV for

mH = 300 GeV. In this example if the mass is 239 GeV for mH = 300 GeV, then the pNGB �5 might

have been observed depending on the magnitudes of its couplings to the other particles.

Another important observation is that either s� or t� should be decreased to make the theory stable

(i.e. where �h(mh1) is positive), such that the mass mH can be pushed up. Thus, the hierarchy problem

in this model is smaller compared to the SM, because the mass of the new fundamental scalar, H, is

smaller �ne-tuned compared to the SM Higgs mass due to its larger mass. The allowed parameter space

will be reduced even further in the next section, where we investigate the vacuum stability of this model.

Page 106 of 193


5.8 The Vacuum Stability

The parameter space investigated in previous section can be constrained even further by studying the

energy where the model becomes unstable, i.e. the energy, when the Higgs self-coupling �h is running to

negative values at a vacuum instability energy E0. The evolution of the self-coupling, �h, is described by

its beta function, which is derived to �rst order in Appendix E. It is given in Eq. (7.95). In the following

calculations we will couple the fundamental Higgs to the SM gauge bosons, such that the beta function

of the self-coupling is (cf. Eq. 3.5 in Ref. [28])

��h � �@�h@�

=1

(4�)2

�24�2h � 6�4t + 12�h�

2t � 3�hg

21 � 9�hg

22 +

3

8

�2g42 + (g21 + g22)

2��; (5.51)

where the �-function of the top-Yukawa coupling, �t, to �rst-loop order including the couplings to the

SM gauge bosons is (cf. Eq. 3.3 in Ref. [28])

��t � �@�t@�

=1

(4�)2

�9

2�3t �

�17

12g21 +

9

4g22 + 8g23

��t

�: (5.52)

The �-function of the SM gauge couplings to �rst-order are (cf. Eq. 3.2 in Ref. [28])

�g1 � �@g1@�

=41

96�2g31 ; �g2 � �

@g2@�

= � 19

96�2g32 ; �g3 � �

@g3@�

= � 7

16�2g33 : (5.53)

The top-Yukawa coupling in the model which is modi�ed with the factor 1= sin� compared to the SM

according to Eq. (7.343), i.e. the top-Yukawa coupling is

�PCHt = �SMt1

sin�: (5.54)

Therefore, the top-Yukawa coupling in this model is equal to or larger than in the SM. In the �-function

of �h the term proportional to �4t will pull it down to negative values. Thus, with a larger top-Yukawa

coupling, the self-coupling will be negative at lower energy than in the SM, i.e. a lower instability energy

E0 compared to the SM (ESM0 � 108 GeV, cf. Figure 2.8).

These �-functions in Eqs. (5.51)-(5.53) are solved for Higgs self-coupling �h as coupled di�erential

equations using Euler's method by Matlab. Matlab is used to calculating the vacuum instability energy

that also calculates the di�erent parameters by using the equations above from the three input parameters

mH , s� and t� . The value of the self-coupling �h(mh1) is calculated at the SM Higgs mass mh1 by using

Eq. (5.46), but for simplicity its running is started at the mass of the top quark mt = 172:44 GeV because

all the other couplings are renormalized at this energy. This is a good approximation because the running

between the mass mh1 = 125:09 GeV and mt = 172:44 is small. The top coupling �t(mt) is calculated

with Eq. (2.90) by inserting the central values in Table 2.3 for mt = 172:44 GeV, mh = 125:09 GeV and

�s(mZ) = 0:1184, which gives �t(mt) = 0:9319.

The vacuum instability energies of the vacuum alignment examples in Table 5.5 are calculated in

Matlab. The values are shown in the table. It can be observed that points in the parameter space can

be very unstable as expected. For example, the vacuum alignment with mH = 1000 GeV, s� = 0:15 and

t� = 1:07 is already unstable at the energies over E0 = 560 GeV though this vacuum alignment gives a

Page 107 of 193


positive self-coupling �h = 0:1041. This is very much below the vacuum instability energy in the SM.

This can again be alleviated by decreasing either mH , s� or t� even more such that the vacuum instability

is increase as observed in Table 5.5. For example, for �xed values, mH = 1000 GeV and t� = 1:07, we

have log10(E0=GeV) = 2:747 for s� = 0:15 and log10(E0=GeV) = 6:738 for s� = 0:15. Therefore, the

parameter space is limited even further by considering the running of the self-coupling.

log10(E

0=GeV

)

mH = 150 GeV mH = 300 GeV

mH = 1000 GeV

log10(E

0=GeV

)

log10(E

0=GeV

)log10(E

0=GeV

)

mH = 5000 GeV

log10(E

0=GeV

)

mH = 104 GeV

log10(E

0=GeV

)

mH = 105 GeV

Figure 5.1: Color plots of the vacuum instability energies E0 as function of sin � and tan� for themasses of the fundamental Higgs mH = 150; 300; 1000; 5000; 104 and 105 GeV calculated and plottedin Matlab. The black plot is where the decay constant is f = vEW. In these calculations it is used thatmt = 172:44 GeV, mh1 = 125:09 GeV and �s(mZ) = 0:1184 from Table 2.3. Therefore g3(mt) = 1:1715and �t(mt) = 0:9319 according to the right panel of Figure 2.6 and Eq. (2.90), respectively.

Page 108 of 193


In Figure 5.1 color plots of the vacuum instability energies E0 are plotted as function of s� and t�

for various masses mH . In these color plots it can be observed that the stable parameter space is getting

smaller when the mass mH increases. Furthermore, the model is most stable in the region where the

angles s� and t� are small. The blue areas in the plots represent models where the self-coupling either is

already negative at EW scale (i.e. the model is unstable) or too large to perturbative calculations (i.e.

��h = �2h=4� > 1). In the last case, we can not say anything about how it is running, it can give a

Landau pole, be unstable or be stable all the way up to the Planck scale, because the perturbation theory

breaks down. To say something about these points we need non-perturbative methods for example lattice

methods.

log10(�h)

�h


Figure 5.2: Color plots of the self-coupling �h(mh1) renormalized at the mass of SM Higgs as function ofsin � and tan� for the mass mH = 300 GeV. Calculated and plotted by Matlab. Left panel: the anglet� goes from 0.1 to 1. Right panel: the angle t� goes from 1 to 15.

In Figure 5.2 the self-coupling �h(mh1) for mH = 300 GeV is plotted for 0:1 < t� < 1 in left panel

and 1 < t� < 15 in right panel, respectively. In upper left panel in Figure 5.3 the plot of the instability

energies for mH = 300 GeV is plotted. The points in the blue region for t� < 1, the self-couplings

�h(mh1) are too large to perturbation theory for points under around t� = 0:8 as shown in left panel in

Figure 5.2. The points in the blue region for t� > 1 have all self-couplings which are negative as shown

in right panel in Figure 5.2, and therefore these models are unstable. Futhermore, there is a narrow strip

in the plot for mH = 150 GeV in Figure 5.1, where the self-couplings are positive and small enough to

use perturbation theory all the way up to the Planck scale.

mH [GeV] 150 300 1000 5000 104 105

smax� (E0 & ESM0 ) 0:25 0:1 0:05 0:01 0:005 0:0005

Table 5.6: The values of the s� angle under which the theory can be nearly equally or more stable thanthe SM in our one-loop approximation (i.e. E0 & ESM

0 � 108 GeV) for various masses mH . The valuesare read o� from Figure 5.1, and therefore these values are approximately values.

Page 109 of 193


If we want a model which is just as good or better than the SM from the vacuum stability point of

view, then the instability energy must be equal to or larger than ESM0 � 108 GeV to �rst-loop order as

shown in Figure 2.8. It can be read o� in the color plots in Figure 5.1 under which maximum s� angle

for all t� we can have an instability energy of the theory which is nearly equal or larger than the value

in SM. These s� angles are shown in Table 5.6 for various mH . For the mass mH = 300 GeV we must

have s� . 0:1 to have a theory which is at least as stable as the SM. However, for mH = 5000 GeV the

s� angle is needed to be below one percent (1=100) and for mH = 105 GeV below one permille (1=1000).

This gives maybe a new �ne-tuning problem of s�, when the mass mH is adjusted up to reduce the the

EW hierarchy problem in SM (vEW=MPlanck � 10�17). In that way the question why the Higgs boson is

so much lighter than the Planck mass (or the grand uni�cation energy or a heavy neutrino mass scale)

is closer to being answered, where the reduction of this hierarchy problem is traded to smallness of the

angle s�.


�V > 0:85�F > 0:75

mH = 300 GeV

m�5 > 500 GeV

mH = 300 GeV

m�5 > 1000 GeV

log10(E

0=GeV

)

log10(E

0=GeV

)

log10(E

0=GeV

)

log10(E

0=GeV

)

Figure 5.3: Color plots of the vacuum instability energies E0 as function of sin � and tan� for the masses ofthe fundamental Higgs mH = 300 GeV with various experimental constraints (no constraints, �V > 0:85and � > 0:75, m�5 > 500 GeV, and m�5 > 1000 GeV respectively). The black plot is where the piondecay constant is f = vEW. Calculated and plotted by Matlab. In these calculations it is used thatmt = 172:44 GeV, mh1 = 125:09 GeV and �s(mZ) = 0:1184 from Table 2.3. Therefore g3(mt) = 1:1715and �t(mt) = 0:9319 according to the right panel of Figure 2.6 and Eq. (2.90), respectively.

Page 110 of 193


In upper left panel in Figure 5.3 the instability energies for mH = 300 GeV are shown without

experimental constraints. In upper right panel the experimental constraints are added for �V and �F in

Eq. (5.50) which from the LHC are at the level of 15% and 25% respectively Ref. [35], i.e. �V > 0:85

and �F > 0:75. It removes some of the points at low t� , but it does not a�ect the results much. These

constraints have also been included in all the plots in Figure 5.1. In the two lower panels the constraints

m�5 > 500 GeV and m�5 > 1000 GeV are added for the mass of the pNGB �5, respectively. For these

constraints some of the stable points in the parameter space are removed, but it has only a small e�ect of

the viability of the model. This constraint has smaller and smaller e�ect for larger masses mH according

to the values in Table 5.5. Therefore, these constraints have nearly no consequences.

The black curves in both Figure 5.1 and Figure 5.3 are where the pion decay constant is f = vEW,

which gives s� as function of t� from the two lowest expressions in table 5.3 as follows

v2EW = f2s2� + v2 = f2s2� + f2t2�s2� ) f2 =

v2EW(1 + t2�)s

2�

;

f = vEW ) s� =1q

1 + t2�

:(5.55)

Below these curves we have the constraint f > vEW, which we want to achieving large enough masses to

the resonances. This constraint restricts not the parameter space more, because the most stable models

are below the curves in Figure 5.1.


We conclude that the parameter space is �ne-tuned in order to obtain a vacuum stable model with a large

mass mH especially the s� angle. The motivation was to reduce or completely remove the electroweak

hierarchy problem in SM. This can only be done at expense of a new �ne-tuning problem. However, a

solution to this �ne-tuning problem could be that the fundamental Higgs mass parametermH is protected

by introducing supersymmetry (SUSY), such that we can have a low mH protected by SUSY and thus

no �ne-tuned s�.

Page 111 of 193

Chapter 6

Conclusions

We began this thesis by considering the potential problems in the SM. Firstly, we can conclude that the

observed Higgs particle at least partially cures violation of unitarity in the weak sector and the custodial

symmetry is minimal broken by the Yukawa sector in the SM. The parameters describing deviation of

the observed boson's coupling to the W bosons and the breaking of the custodial symmetry are measured

to be �W = 0:91+0:10�0:12 and � = 1:0006� 0:0009 at the LHC and LEP experiments, respectively. They are

normalized such that they are both one in the SM. If �W was one then the observed boson would be fully

responsible for unitarizingWLWL scattering as shown in Eq. (2.45). According to experimental data from

LEP experiments, the EW precision parameters are measured to be S = 0:05� 0:10 and T = 0:08� 0:12

(the � parameter is related to T ), which are normalized to be zero in the SM. Therefore, the SM are

consistent with the measurements of these parameters. We have shown perturbatively that although

in isolation the SM Higgs sector is trivial this is modi�ed when the top-Yukawa coupling is included.

Instead, the SM is possibly vacuum unstable, because the Higgs self-coupling, �, becomes negative at

energies above its instability energy, E0 � 108 GeV, as computed in the one-loop approximation.

Despite of all the successes of the SM it cannot explain all observations and there are a various

reasons that it cannot be the ultimate theory of Nature. This includes the existence of neutrino masses,

baryogenesis, dark matter and dark energy. An important reason to believe the SM is not a complete

description of EW symmetry breaking is that the Higgs boson is unnatural, because its bare mass must

be very �ne-tuned to an absurd precision of about 1 : 1032 (according to the BG quantity in Eq. (2.117))

to achieve the correct physical mass.

This naturalness problem is addressed in composite formulations of the Higgs mechanism. By intro-

ducing technicolor the Higgs mechanism has a natural dynamical origin and is simultaneously non-trivial

in analogy with chiral symmetry breaking in QCD. It does not explain the origin of SM fermion masses,

and therefore we introduced extended technicolor to explain these masses. Such ETC models cause their

own set of problems. It is challenging to generate enough mass to the heaviest fermions (in some realiza-

tions it is already problematic to produce the mass of the charm quark), ETC contributes to the �avor

changing neutral currents (FCNC) and contributes to discrepancies with precision electroweak measure-

112

CHAPTER 6. CONCLUSIONS

ments. We discussed that these potential issues can be alleviated by assuming that the gauge coupling

constant of TC evolves slowly between the scales �TC and �ETC. These kind of models are called walking

TC models. However, the potential problem with these TC models is that the Higgs boson is needed

to be identi�ed with the resonance techni-�, which seems to be too heavy to play the role as the Higgs

boson although see e.g. Ref. [70] for a discussion of how top-quark corrections may change this.

This problem is alleviated in CH models, where the vacuum is aligned away from the TC vacuum. In

these models we identi�ed the Higgs boson as one of the pNGBs, h, with the desired small mass. The

next lightest resonance, pNGB �, has also large enough mass to explain why we have not observed it yet.

Unfortunately, these models have another issue, which is that the top-loop and the explicit mass term

contributions to the Higgs potential seems to be �ne-tuned compared to each other.

Thus, we investigated the possibility with a PCH model, where the Higgs boson is partially composite

and fundamental. By a novel analysis of the vacuum stability of this model, we conclude that the

parameter space is needed to be �ne-tuned to obtain a vacuum stable model with a large fundamental

mass parameter mH . We had the motivation to reduce or completely remove the EW hierarchy problem

in the Higgs sector, but this can only be done at expense of a possibly new �ne-tuning problem. However,

this model may be saved by introducing supersymmetry, such that we can have a low mass parameter

mH protected by supersymmetry and thus no �ne-tuned s�.

Page 113 of 193

Chapter 7

Appendices

Appendix A: SU(4) generators

In this appendix we have written the generator matrices of the SU(4) group. It is convenient to use the

following representation of SU(4)

Sa =

0@ A B

By �AT

1A ; a = 1; : : : ; 6 and Xi =

0@ C D

Dy CT

1A ; i = 1; : : : ; 9 (7.1)

where A is hermitian, B = �BT , C is hermitian and traceless and D = DT . The �rst four matrices are

Sa =1

2p2

0@ �a 0

0 ��aT

1A ; a = 1; : : : ; 4; (7.2)

where a = 1; : : : ; 3 are the Pauli matrices,

�1 =

0@ 0 1

1 0

1A ; �2 =

0@ 0 �i

i 0

1A and �3 =

0@ 1 0

0 �1

1A ; (7.3)

and �4 = I. The next two matrices are

Sa =1

2p2

0@ 0 Ba

Bay 0

1A ; a = 5; 6; (7.4)

where B5 = �2 and B6 = i�2. The matrices Sa are the generators of the SO(4) group. The last nine

matrices are

Xi =1

2p2

0@ � i 0

0 � iT

1A ; i = 1; 2; 3; (7.5)

Xi =1

2p2

0@ 0 Di

Diy 0

1A ; i = 4; : : : ; 9; (7.6)

114

CHAPTER 7. APPENDICES

where D4 = I, D5 = iI, D6 = �3, D7 = i�3, D8 = �1 and D9 = i�1. The ten generator matrices of

symplectic group Sp(4) are Sa with a = 1; : : : ; 4 and Xi with i = 4; : : : ; 9. These generators satisfy the

following commutation relations (Eq. (2.10) in Ref. [16])

[Sa; Sb] = ifabcSc

[Xa; Sb] = ifabcXc

[Xa; Xb] = ifabcSc

(7.7)

From the second commutation relation, we have that

[Xa; Sb] = ifabcXc ) XaSb � SbXa = ifabcXc )E�XaE�E�SbE� � E�SbE�E�XaE� = ifabcE�XcE�;

(7.8)

where E� are the vacuum matrices for SO(4) and Sp(4), respectively, given in Eq. (3.13), and if it is

true that

SaTE� + E�Sa = 0) SaT = �E�SaE�;XaTE� � E�Xa = 0) XaT = E�XaE�;

(7.9)

then

E�XaE�E�SbE� � E�SbE�E�XaE� = ifabcE�XcE� )�XaTSbT + SbTXaT = ifabcXcT )� SbXa +XaSb = ifabcXc ) [Xa; Sb] = ifabcXc:

(7.10)

Therefore, we have that

SaTE� + E�Sa = 0;

XaTE� � E�Xa = 0:(7.11)

Rotation of the Sp(4) Generators Matrices into a General Vacuum

In the following the Sp(4) and SU(4)=Sp(4) generators will be written with a general Sp(4) metric

� =

0@ ei� cos �� sin �12

� sin �12 �e�i� cos ��

1A ; (7.12)

which is the general vacuum (derived in Appendix C-4). We can write an arbitrary SU(N) generator as

T = Tk + T?, which satisfy the relations

Tk�+ �TTk = 0;

T?�� TT? = 0;(7.13)

where Tk and T? are the projection of the SU(N) generators T on parallel with the Sp(4) generators and

perpendicular to Sp(4) generators (i.e. the SU(N)=Sp(N) generators), respectively. These are projections

in the sense that

(Tk)? = 0;

(Tk)k = Tk:(7.14)

Page 115 of 193


If these projections satisfy these condition, then we have

Tk�+ �TTk = 0) Tk +�TT?�y = 0;

T?�� TT? = 0) T? � �TT?�y = 0

(7.15)

together with (Tk)? = 0 and (Tk)k = Tk give the projection generators

Tk =1

2(T � �TT�y);

T? =1

2(T +�TT�y):

(7.16)

To derive the form of the generators we will start with the EW vacuum in the composite limit

�0 =

0@ � 0

0 ��

1A ; (7.17)

and by performing a SU(4) rotation we obtain the general vacuum � = U0�0UT0 , where

�0 =

0@ cos �212 sin �

2 �

� sin �2 � cos �212

1A 2 SU(4): (7.18)

By performing the rotation of the generators T = U0T0Uy0 , we obtain the generators in a general vacuum,

because

Tk;0�0 +�0TTk;0 = 0) U0Tk;0U

y0U0�0U

T0 + U0�0U

T0 U

�0T

Tk;0U

T0 = 0) Tk�+ �TTk = 0; (7.19)

where � = U0�0UT0 and Tk = U0Tk;0U

y0 , and in the same way with the broken generators T? = U0T?;0U

y0 .

Thus by performing the rotation of the generators T = U0T0Uy0 , we obtain the broken generators in a

general vacuum are given by

T 1? =

1

2p2

0@ s��1 �c��3�c��3 s��1

1A ; T 2

? =1

2p2

0@ s��2 ic�12

�ic�12 �s��2

1A ; T 3

? =1

2p2

0@ s��3 c��1

c��1 s��3

1A ;

T 4? =

1

2p2

0@ 0 �2

�2 0

1A ; T 5

? =1

2p2

0@ c�12 �s�

s�� c�12

1A ;

(7.20)

and the unbroken generators are given by

T 1k =

1

2p2

0@ �1 0

0 ��1

1A ; T 2

k =1

2p2

0@ �2 0

0 �2

1A ; T 3

k =1

2p2

0@ �3 0

0 ��3

1A ; T 4

k =1

2p2

0@ c��1 s��3

s��3 c��1

1A ;

T 5k =

1

2p2

0@ c��2 �is�12

is�12 �c��2

1A ; T 6

k =1

2p2

0@ c��3 �s��1�s��1 c��3

1A ; T 7

k =1

2p2

0@ 0 i�1

�i�1 0

1A ;

T 8k =

1

2p2

0@ s�12 c��

�c�� s�12

1A ; T 9

k =1

2p2

0@ 0 i�3

�i�3 0

1A ; T 10

k =1

2p2

0@ 0 12

12 0

1A :

(7.21)

Page 116 of 193


Appendix B: Gauge Anomaly Cancellation

In this section, we will show that the Standard Model with a gravitational interaction and the Minimal

Walking Technicolor model are gauge anomaly free. In a gauge theory in which gauge bosons couple

to a chiral current, the triangle diagrams appear in the one-loop corrections to the three-gauge-boson

vertex function (see Figure 7.1. These anomalous terms violate the Ward identity for this amplitude.

The theories can be gauge invariant only if these anomalous contributions disappear.

p3; � 5

p1; �

p2; �

k p3; � 5

p1; �

p2; �

k

Figure 7.1: Feynman diagrams for the triangle anomaly

The simplest Green function where the anomalies occur is the three-point function of two vector and one

axial-vector currents,

T ijk��(x; y; z) = hTji�(x)jj�(y)jk� (z)i; (7.22)

where i; j; and k can take the values V;A; and P , which require to replace the j by the vector current

ja� = �Ta , the axial-vector current j5a� = 5 �T

a , and the energy-momentum four-vector pa�,

respectively. The corresponding Ward identities for a local chiral transformation

0 = ei�(x) 5 (x)

0= ei�(x) 5

(7.23)

can be calculated (shown in Eq. (2.7.16) in Ref. [13]), which yields the expressions

@�xTV V A�� (x; y; z) = @�yT

V V A�� (x; y; z) = 0

@�zTV V A�� (x; y; z) = 2mTV V P�� (x; y; z):

(7.24)

This is what should happen, when there would be no anomalies.

Now, we will calculate the triangle-graph anomaly in Figure 7.1. The two diagrams for TV V A�� are

UV-divergent. Therefore, they must be regularized. This can be done by Pauli-Villars regularization by

subtracting the same diagrams with mass M � m. By using the Feynman rules we get

TV V A�� (p1; p2; p3 = �p1 � p2) = �i3�

d4k

(2�)4�Tr� �(=k �m)�1 �(=k � =p2 �m)�1 � 5(=k + =p1 �m)�1

�+

Tr� �(=k �m)�1 �(=k � =p1 �m)�1 � 5(=k + =p2 �m)�1

�� (m!M)

�:

(7.25)

Now, the integral is �nite. In order to test the the �rst two Ward identities in Eq. (7.24) we multiply

Page 117 of 193


with p�1 or p�2 , and decompose =p1 = (=k + =p1 �m)� (=k �m) = �(=k � =p1 �m) + (=k �m), which gives

p�1TV V A�� (p1; p2; p3 = �p1 � p2) = �i

�d4k

(2�)4�Tr�� (=k �m)�1 �(=k � =p2 �m)�1 � 5 + (=k + =p1 �m)�1 �(=k � =p2 �m)�1 � 5

�+

Tr�(=k + =p2 �m)�1 �(=k �m)�1 � 5 � (=k + =p2 �m)�1 �(=k � =p1 �m)�1 � 5

�� (m!M)

�:

(7.26)

By performing the shifts k ! k + p2 and k ! k + p2 � p1, we �nd that the integrand vanishes. The

same can be shown by multiplying the integral with p�2 . Consequently, the two vector Ward identities

are ful�lled.

The axial-vector case can be studied by multiplying the three-point function in Eq. (7.25) with p�3

and decompose =p3 5 = �(=p1+=p2) 5 = (=k�=p2�m) 5+ 5(=k+=p1�m)+2m 5 = (=k�=p1�m) 5+ 5(=k+

=p2 �m) + 2m 5. We get that

p�3TV V A�� (p1; p2; p3 = �p1 � p2) = 2i

�d4k

(2�)4�mTr

� �(=k �m)�1 �(=k � =p2 �m)�1 5(=k + =p1 �m)�1

�+

mTr� �(=k �m)�1 �(=k � =p1 �m)�1 5(=k + =p2 �m)�1

��

MTr� �(=k �M)�1 �(=k � =p2 �M)�1 5(=k + =p1 �M)�1

��

MTr� �(=k �M)�1 �(=k � =p1 �M)�1 5(=k + =p2 �M)�1

��=

2mTV V P�� +A�� ;

(7.27)

where by replacing � 5 by 5 in the diagrams in Figure 7.1 we get

TV V P�� = i

�d4k

(2�)4

�Tr� �(=k �m)�1 �(=k � =p2 �m)�1 5(=k + =p1 �m)�1

�+

Tr� �(=k �m)�1 �(=k � =p1 �m)�1 5(=k + =p2 �m)�1

��;

(7.28)

and the integral (calculated on page 274 in Ref. [13])

A�� =� limM!12iM

�d4k

(2�)4

�Tr[ �(=k +M) �(=k � =p2 +M) 5(=k + =p1 +M)]

(k2 �M2)[(k � p2)2 �M2][(k + p1)2 �M2]+

(�; 1)! (�; 2)

�= limM!116M2"��p

�1p�2

i

16�2�12M2

= � i

2�2"��p

�1p�2 :

(7.29)

This integral is UV-�nite but non-vanishing. Therefore, we have an anomaly. The modi�ed (anomalous)

Ward identities for the regularized one-loop vertex function are

p�1TV V A�� = p�2T

V V A�� = 0

p�3TV V A�� + 2mTV V P�� i

2�2"��p

�1p�2 :

(7.30)

Thus the vector currents are anomaly-free whereas the axial-vector current has an anomaly from quantum

�uctuations. In the case of non-abelian currents, the coupling matrices T a enter. Generally, the triangle-

Page 118 of 193


graph anomaly of the axial-vector current is

Aabc�� =Tr[fT a; T bgT c]

2

�i2�2

"��p�1p�2

(7.31)

Therefore, the anomalous term of a triangle diagram of three gauge bosons is proportional to

Tr[ 5T afT b; T cg] = Tr[T aLfT bL; T cLg]� Tr[T aRfT bR; T cRg]; (7.32)

where the trace is over all fermion species. The factor 5 is associated with chiral currents. This factor

is equal to �1 for left-handed fermions and +1 for right-handed fermions. The anticommutator comes

from that we take the sum of the two di�erent triangle diagrams in which the fermions circle in opposite

directions in Figure 7.1.

Now, we will show that the Standard Model (with symmetry group SU(3) SU(2) U(1)) and a

model of the gravitational force is anomaly free. We can omit the diagrams with three SU(3) bosons

or of one SU(3) and two gravitons because all of the couplings are left-right symmetric. The full set of

diagrams is shown in Figure 7.2.

U(1)

U(1)

U(1) U(1)

SU(2)

U(1)

U(1)

SU(2)

SU(2)

U(1)

SU(3)

U(1)

U(1)

SU(3)

SU(2)

SU(3)

SU(3)

U(1) SU(2)

SU(2)

SU(2)

SU(2)

SU(2)

SU(3)

SU(2)

SU(3)

SU(3)

Grav.

Grav.

U(1)

Figure 7.2: Possible gauge anomalies of the SM and a model of a graviton. All of these anomalies mustvanish for the theory to be consistent.

We have that the anomaly of three SU(2) gauge bosons always vanishes because of the property of Pauli

matrices f�a; � bg = 2�ab. There we have that

Tr[�af� b; � cg] = 2�abTr[�a] = 0; (7.33)

and therefore the anomaly vanishes. The anomalies where the diagram is containing one SU(2) or SU(3)

gauge boson will always vanish, because they are proportional to Tr[�a] = 0 or Tr[�a] = 0 where �a and

�a are Pauli and Gell-Mann matrices respectively.

The remaining nontrivial anomaly diagrams are shown in Figure 7.3. The anomaly in the upper left panel

with three U(1) gauge bosons is proportional to

Tr�Y 3�= 3

h2�� 1

6

�3+�23

�3+�� 1

3

�3i� 2�� 1

2

�3+ (�1)3 = 0; (7.34)

Page 119 of 193


where the sum involving both left- and right-handed quarks and leptons with an extra -1 for the left-

handed particles. The factor 3 counts the three color states of the quarks.

U(1)

U(1)

U(1) U(1)

U(1) U(1)

SU(3)

SU(3)

Grav.

Grav.

SU(2)

SU(2)

Figure 7.3: The remaining nontrivial gauge anomalies.

The anomaly in the upper right panel with two SU(2) bosons and one U(1) boson is proportional to

Tr[�a� bY ] =1

2�abXfL

YfL =1

2�ab��3 � 16 � �� 1

2

��= 0; (7.35)

where the sum runs over the left-handed fermions. The anomaly in the lower left panel with two SU(3)

bosons and one U(1) boson is proportional to

Tr[�a�bY ] =1

2�abXq

Yq =1

2�ab � 3 ��2 � 16 + 2

3 +�� 1

3

��= 0; (7.36)

where the sum is over the left-handed and right-handed quarks. The anomaly with two gravitons and

one U(1) gauge boson is proportional to

Tr[Y ] = 3��2 � 16 + 2

3 +�� 1

3

�� 2�� 1

2

�+ (�1) = 0: (7.37)

Therefore the Glashow-Weinberg-Salam theory is completely free of axial vector anomalies among the

gauge currents.

An extended gauge symmetry group of the SM model G = SU(3)CSU(2)TCSU(2)LU(1)Y with

the technicolor symmetry group SU(2)TC. This theory is called the Minimal Walking Technicolor theory.

The gauge anomalies cancel with the following generic hypercharge assignment

Y (QL) =y

2; Y (UR; DR) =

�y + 1

2;y � 1

2

�;

Y (LL) = �3y2; Y (NR; ER) =

��3y + 1

2;�3y � 1

2

�;

(7.38)

Page 120 of 193


where the parameter y can take any real values. We recover the SM hypercharges for y = 1=3, and the

electric charge is Q = T 3 + Y , where T 3 is the weak isospin generator.

We must check that the theory is gauge anomaly free. The anomaly with three U(1) gauge bosons as in

Eq. (7.34) is proportional to

Tr�Y 3�=3h�2 �y2�3 + �y+12 �3 + �y�12 �3i� 2

��3 � y2�3 + ��3y+12

�3+��3y�1

2

�3=1

8(y2 + y + 1 + 2y2 + 2y � y2 + y � 1� 2y2 + 2y + 9y2 � 3y + 1 + 18y2�

6y � 9y2 � 3y � 1� 18y2 + 6y) = 0;

(7.39)

where the hypercharges in Eq. (7.38) have been used. This can be done for the anomaly with two SU(2)

bosons and one U(1) boson as in Eq. (7.35) which is proportional to

Tr��a� bY

�=

1

2�abXfL

YfL =1

2�ab��3y2 � ��3y2�� = 0: (7.40)

For the anomaly with two SU(3) bosons and one U(1) boson as in Eq. (7.36) is proportional to

Tr��a�bY

�=

1

2�abXq

Yq =1

2�ab � 3 ��2y2 + y+1

2 + y�12

�= 0; (7.41)

and for the anomaly with two gravitions and one U(1) boson as in Eq. (7.37) is proportional to

Tr [Y ] = 3��2y2 + y+1

2 + y�12

�� 2��3y2�+ �3y+1

2 + �3y�12 = 0: (7.42)

Therefore the gauge anomalies is cancelled in the Minimal Walking Technicolor theory with the hyper-

charge assignment in Eq. (7.38).

Page 121 of 193


Appendix C: Group Representations

We have been given the structure coe�cients fabc of a nonabelian group. The representation of that

group R is speci�ed by a set of D(R) D(R) traceless hermitian matrices T aR that the commutation

relations

[T aR; TbR] = ifabcT cR (7.43)

where D(R) is the dimension of the representation and this commutation relations are the same as the

original generators matrices T a. These original generator matrices T a's corresponds to the fundamental

representation.

If we have a unitary transformation,

V �1TiV = �(Ti)�; (7.44)

then for V = I such that Ti = �(Ti)� for every i, we have that the representation R is real. If V 6= I, we


is complex.

In this case, the complex conjugate representation �R is speci�ed by

T a�R = �(T aR)� (7.45)

It can be shown that the matrix V is only unique up to a constant �. We have from Eq. (7.44) that

V T �V �1 = �T and QT �Q�1 = �T )V T �V �1 = QT �Q�1 ) T �V �1 = V �1QT �Q�1 ) V �1QT � = T �V �1Q) [V �1Q;T �] = 0) (7.46)

V �1Q = �1) Q = �V

From line two to line three in Eq. (7.46) we have used that V �1Q must be a constant of the unit matrix

according to Schur's �rst lemma. Therefore we have that the matrix V is only unique up to a constant

�.

It can also be shown that the generator matrices have eigenvalue +1 for if the representation R is real

and -1 for if it is pseudoreal. We have according to Eq. (7.44) that

V T �V �1 = �T ) TTV �1 = �V �1T (7.47)

We can use Eq. (7.44) and Schur's �rst lemma to get that

V T �V �1 = �T ) V TT = �TV ) TV T = �V TTT ) TV TV �1 = �V TTTV �1 )TV TV �1 = V TV �1T ) TV TV �1 � V TV �1T = 0) [T; V TV �1] = 0) V TV �1 = a1) (7.48)

V T = aV ) V = aV T = a2V ) a = �1;

where a = 1 and a = �1 are the eigenvalues for a real and pseudoreal representation respectively. If least

Page 122 of 193


one generator matrix T aR (or a real linear combination of them) has eigenvalues that is not a = �1 then

the representation is complex.

Another important representation of the compact nonabelian group is the adjoint representation A

which is given by

(T aA)bc = �ifabc (7.49)

The structure constants fabc are real and therefore the generator matrices satisfy the condition T aA =

�(T aA)�. Thus the adjoint representation is real.

We have that the complex conjugate transforms as follows

U !gU = ei�aTaRU )

�U ! �UAgyAB = �UA(g

�)TAB = g�BA �UA = e�i�aTa�R �U = ei�

a(�Ta�R ) �U = ei�aTa�R �U: (7.50)

Thus, we have that

U !gU = ei�aTaRU and (7.51)

�U !g� �U = ei�aTa�RU; (7.52)

where the generators in the complex conjugate representation are T �R = �T �R. From the algebra of the

representation of the group R we obtain that

[T aR; TbR] = ifabcT cR )

[�T a�R ;�T b�R ] = �ifabcT c�R ) (7.53)

[T a�R; Tb�R] = ifabcT c�R;

which de�nes that complex conjugate representations have the same algebra, i.e. that the generators T a�R

ful�l the same commutation relations.

Page 123 of 193


Appendix D: Goldstone Theorem

In this section we will determine the consequence by breaking symmetries at the quantum level. The

consequence is described by the Goldstone theorem. The Goldstone theorem states the following: If a

symmetry group G of size dim G is broken, then there exists as many massless particles as there are

generators. If the group is only broken partly than only as many massless particles appear as generators

are broken.

To determine the Goldstone theorem at quantum level, it is useful to investigate the normalized

partition function

T [Ji] =Z[Ji]

Z[0]=

1

Z(0)

�D�i exp

�i

�d4x(L+ Ji�i)

�; (7.54)

where L is the Lagrangian of the theory, Ji are the sources of the �elds �i. The variation of the partition

function is

0 = �Z[Ji] =

�D�i exp

�i

�d4x(L+ Ji�i)

��d4x

�@��i@�j

+ �

�iS + i

�d4xJi�i

��; (7.55)

which is vanishing, because the Lagrangian and the measure are invariant under a symmetry transfor-

mation. The �rst term is the variation of the measure which is invariant, and therefore it vanishes. The

second term is the variation of the action, which also vanishes. The third term must also vanish, therefore

we have that �d4xJiT

aik

�T [Ji]

i�Jk= 0; (7.56)

where it has been used that Z[0] is constant, and

�T [Ji]

i�Ji=

1

Z[0]

�D�i�i exp

�i

�d4x(L+ Ji�i)

�: (7.57)

The relation between the generating functional of Green functions T [Ji] and generating functional of

connected Green functions Tc[Ji] is

T � eTc ) �T = �(eTc) = eTc�Tc (7.58)

By inserting Eq. (7.58) into Eq. (7.56) we get Eq. (7.56) in terms of the generating functional for

connected Green functions

eTc�d4xJiT

aik

�Tc[Ji]

i�Jk= 0: (7.59)

This can be transformed into an equation in terms of the generating functional of vertex functions �.

This is related to connected one by a Legendre transformation

i�[�] =� i�d4xJi�i + Tc[J ];

h�ii =h0j�i[Ji]j0i = �Tc[J ]

i�Ji;

Ji =� ��[�]

i��i:

(7.60)

Page 124 of 193


By exchanging the derivative and the source we get that

�d4x

��

��iT aikh�ki = 0: (7.61)

When the �elds �i developing a vacuum expectation value vi, it then holds

vi =h�ii = �Tci�Ji

[0]

Ji =� ��

i��i[vi]

(7.62)

By di�erentiating Eq. (7.61) with respect to the �eld we get the equation

�d4x

��2�

��i(x)�j(y)T aikh�ki+

��

��iT aii�(x� y)

�; (7.63)

where the last term vanishes since the generators are traceless or because i��=��ij�i=vi = Ji = 0. If we

use the inverse propagator of the �elds �i

i�2�

��i(x)��j(y)[vi] = �(D�1)ik(x� y); (7.64)

and that the �rst term in Eq. (7.63) is just the Fourier-transform of the inverse propagator at zero

momentum, which yielding

(G�1)ij(p = 0)T aikvk = 0: (7.65)

Thus, there must vanish as many inverse propagators as there are non-zero vi. The inverse propagator

at tree-level is

(G�1)ij = �ij(p2 +m2); (7.66)

which implies that the pole mass must vanish, and therefore the propagator becomes a propagator of a

massless particle.

This con�rmes the Goldstone theorem: If you have a symmetry group G which breaks to the subgroup

H (called the stability group of G), i.e. that the coset space G=H of size dim G=H is broken, because

the �elds �i get vacuum expectation values vi. Thus, there will exist as many massless particles as there

are broken generators, i.e. dim G=H generators.

Page 125 of 193


Appendix E: Beta Functions

In this section, we will calculate the �-function for the Higgs four-point self-coupling �. To calculating

the �-function, we need the Feynman rules of these Lagrangian terms

�

4(�y�)2 � �t�abQ�La��btR � �t�baQLa�bt�R

=�

16(�21 + �22 + �23 + (v + h)2)2 � �tp

2

�t�L(v + h+ i�3)tR � tL(v + h� i�3)t�R

�+ : : :

=�

16h4 +

�

8(�21 + �22 + �23)h

2 � �tp2h(t�LtR + t�RtL) + : : :

=�

16h4 +

�

8(�21 + �22 + �23)h

2 � �tp2h�tt+ : : : ;

(7.67)

where the complex Higgs doublet is written

�(x) =1p2

0@ �2(x) + i�1(x)

v + h(x)� i�3(x)

1A : (7.68)

To determine the �-function of a coupling constant, we need to look at the relation between the bare

�0 and renormalized coupling � of both Higgs four-point coupling and the Yukawa coupling to the top

quark, which are

�0 = Z�2� Z�~��; �t0 = Z

�1=2� Z�1 Z�t ~�

�=2�t; (7.69)

which is renormalized using MS scheme with �2 = 4�e� ~�2. The constants Z�, Z�, Z and Z�t are

the scalar wave function, the Higgs four-point coupling, spinor wave function and top-Yukawa coupling

renormalization constant, respectively. We take the logarithm of these relations on both sides, which

gives

ln�0 =

1Xn=1

Ln(�t; �)

�n+ ln�+ � ln ~�;

ln�t0 =

1Xn=1

Gn(�t; �)

�n+ ln�t +

1

2� ln ~�;

(7.70)

where we have de�ned

ln(Z�2� Z�) �1Xn=1

Ln(�t; �)

�n;

ln(Z�1=2� Z�1 Z�t) �

1Xn=1

Gn(�t; �)

�n:

(7.71)

Thereafter, we di�erentiate the two equation in Eq. (7.70) with respect to ln� and multiply with � and

�t on both sides, respectively, which gives us

0 =

1Xn=1

��@Ln@�t

d�td ln�

+ �@Ln@�

d�

d ln�

�1

�n+

d�

d ln�+ ��;

0 =

1Xn=1

��t@Gn@�t

d�td ln�

+ �t@Gn@�

d�

d ln�

�1

�n+

d�td ln�

+1

2��t:

(7.72)

In a renormalizable theory, we have that d�=d ln� and d�t=d ln� must be �nite when �! 0. Therefore,

Page 126 of 193


we can write the two equations above as follows

d�td ln�

= �1

2��t + ��t(�t; �);

d�

d ln�= ��+ ��(�t; �):

(7.73)

By inserting the two expression into the two equations in Eq. (7.72) we �nd that the �-function of the

two couplings are

��(�t; �) = �

�1

2�t

@

@�t+ �

@

@�

�L1;

��t(�t; �) = �t

�1

2�t

@

@�t+ �

@

@�

�G1;

(7.74)

where the coe�cients to higher orders of 1=� must vanish. These expressions can be used to calculate the

�-functions, which according to Eq. (7.73) encodes how the couplings develop when the energy scale �

changes.

We will start to calculate the amplitudes of one-loop correction to the Higgs four-point vertex with

a scalar loop as shown in Figure 7.4. The scalar in the loop is either the Higgs (h) or one of the three

would-be Goldstone bosons (�1;2;3). There is such an one-loop correction in both s-, t- and u-channel,

which each gives identical contribution to the �-function. In addition, the diagrams have also a symmetry

factor of 1=2, which we multiply on the amplitude.

q + p

p

pp

s-channel t-channel u-channel

Figure 7.4: The three diagrams for the one-loop correction to the Higgs four-point vertex with a scalarloop, where there is either Higgs itself or one of the three would-be Goldstone bosons in the loop.

The amplitude of the these diagrams is calculated as follows

iM4;scalar�loop

= 31

2

��i4! �16�2 + 3��i4�8 �2�

�d4q

(2�)4i

q2 �m2 + i�

i

(q + p)2 �m2 + i�

=9

2�2

� 1

0

dx1dx2

�d4q

(2�)41

(x1(q2 �m2 + i�) + x2((q + p)2 �m2 + i�))2�(x1 + x2 � 1)

=9

2�2

� 1

0

dx2

�d4q

(2�)41

(q2 �m2 + i�+ x2(p2 + 2qp))2

=9

2�2

� 1

0

dx

�d4l

(2�)41

(l2 ��)2=

9

2�2

� 1

0

dxi

(4�)d=2�(2� d

2 )

�(2)

�1

�

�2�d2 d=4��!

(7.75)

Page 127 of 193


9

2�2

i

(4�)2

� 1

0

dx

�2

�� log�� + log(4�) +O(�)

�

=9i

2(4�)2�2

� 1

0

dx

�2

�� log(m2 � x(1� x)p2)� + log(4�) +O(�)

�

=9i

2(4�)2�2�2

�+ �nite

�;

where we have used dimensional regularization to regularize the loop integral, and we have de�ned

� � m2 � x(1� x)p2

l � q + xp) l2 = q2 + 2xpq + x2p2:(7.76)

The next diagram, we will calculate to the one-loop correction to the Higgs four-point vertex, is the

diagram in Figure 7.5 with a top-loop and four external Higgs lines. There are �ve other permutations

of this diagram, which are obtained by permuting the external momenta. In addition, the amplitude of

these diagrams must be multiplied by 3 because of the color charge of the top quark in the loop. We get

also a factor �1, because we have a fermion loop.

q1

q4 q2

q3

p1

p2

p3

p4

Figure 7.5: One of six diagrams with a top-loop and four external Higgs lines. The other �ve areobtained by permuting the external momenta.

The amplitude of these diagrams is

iM4;top�loop = 6 � (�1) � 3(�i �tp2)4

�d4q

(2�)4Tr[(=q +m)(=q � =p1 +m)(=q � =p1 � =p2 +m)(=q � =p3 +m)]

(q2 �m2 + i�)((q � p1)2 �m2 + i�)((q � p1 � p2)2 �m2 + i�)((q � p3)2 �m2 + i�);

(7.77)

where we have used the four-momentum conservation in the vertices to derive the four-momenta in the

loop, which are

p1 � q1 + q4 = 0; p2 � q4 + q3 = 0; � p3 + q1 � q2 = 0; and � p4 + q1 � q3 = 0)q1 = q; q2 = q � p3; q3 = q � p1 � p2; and q4 = q � p1:

(7.78)

We can rewrite the denominator by introducing Feynman parameters as follows

1

(q2 + i�)((q � p1)2 + i�)((q � p1 � p2)2 + i�)((q � p3)2 + i�)=

� 1

0

dx1dx2dx3dx4

6�x1(q2 + i�) + x2((q � p1)2 + i�) + x3((q � p1 � p2)2 + i�) + x4((q � p3)2 + i�)

�4

Page 128 of 193


�(x1 + x2 + x3 + x4 � 1) = 6

� 1

0

dx2dx3dx4

�q2 �m2 + i�+ x2(p

21 � 2q � p1)+

x3(p21 + p22 � 2(q � p1 + q � p2 � p1 � p2)) + x4(p

23 � 2q � p3)

��4= 6

� 1

0

dx2dx3dx4�q2 �m2 + i�+ x2p

21 + x3(p1 + p2)

2 + x4p23 � 2(x2p1 + x3(p1 + p2) + x4p3)q

��4=

6

� 1

0

dx2dx3dx4

�l2 � (x2p1 + x3(p1 + p2) + x4p3)

2 + 2x3p1 � p2 + i��4

=

6

� 1

0

dx2dx3dx41

(l2 ��)4;

(7.79)


l � q � (x2p1 + x3(p1 + p2) + x4p3))l2 = q2 + (x2p1 + x3(p1 + p2) + x4p3)

2 � 2(x2p1 + x3(p1 + p2) + x4p3)q

� � (x2p1 + x3(p1 + p2) + x4p3)2 � 2x3p1 � p2 � i�:

(7.80)

The momentum q can be written to di�erent power as

l = q � (x2p1 + x3(p1 + p2) + x4p3))q = l + x2p1 + x3(p1 + p2) + x4p3 = l + � )q2 = l2 + �2 + 2l� )q4 = l4 + 4l3� + 6l2�2 + 4l�3 + �4;

(7.81)

which can be used to rewrite the numerator in the following. The numerator can be written as

Tr�=q(=q � =p1)(=q � =p1 � =p2)(=q � =p3)

�= 4h(q2)2 � q2q � p3 + q2q � p3 � q � p3q2 � q2p1 � q+

q � p1q2 � q2q � p1 + q2p1 � p3 � q � p1q � p3 + q � p3q � p1 � q2p2 � q + q � p2q2 � q2q � p2+q2p2 � p3 � q � p2q � p3 + q � p3q � p2 � q � p1q2 + q2p1 � q � q2p1 � q + q � p1q � p3 � q2p1 � p3+q � p3p1 � q + q � p1p1 � q � q � p1p1 � q + q2p21 � q � p1p1 � p3 + q � p1p1 � p3 � q � p3p21 + q � p1p2 � q�q � p2p1 � q + q2p1 � p2 � q � p1p2 � p3 + q � p2p1 � p3 � q � p3p1 � p2

i=

4hq4 � q � p3q2 � 2q2q � p1 + q2p1 � p3 � q2p2 � q + q2p2 � p3 � q � p1 � q2 + 2q � p1q � p3 � q2p1p3+

q2p1 � p2 � q � p1p2 � p3 + q � p2p1 � p3 � q � p3p1 � p2i=

4�l4 + l2(6�2 � � � p3 � 2� � p1 � � � p2 + p2 � p3 + p1 � p2) + l�l�(�2p3�� 4��p1� � 2��p2�+

2p1�p3� + �4 � � � p3�2 � �2� � (2p1 + p2) + �2p2 � p3 + 2� � p1� � p3 + �2p1 � p2 � � � p1p2 � p3�� p2p1 � p3 � � � p3p1 � p2

�= 4�l4 + l2A+ l�l�B�� + C

�

(7.82)

By inserting the denominator and numerator we get the amplitude

iM4;top�loop

= �6 � 7214�4t

�dx2dx3dx4

�ddl

(2�)dl4 + l2A+ l�l�B�� + C

(l2 ��+ i�)4

Page 129 of 193


= �6 � 18i�4t� 1

0

dx2dx3dx41

(4�)d=2

"d(d+ 2)

4

��2� d

2

��(4)

�1

�

�2�d2�Ad

2

��3� d

2

��(4)

�1

�

�3�d2

�B�� g��

2

��3� d

2

��(4)

�1

�

�3�d2+ C

��4� d

2

��(4)

�1

�

�4�d2#

d=4��!

� 6 � i18�4t� 1

0

dx2dx3dx41

(4�)2

"�2

�� log�� + log(4�) +O(�)

�� 2A

1

��B�� g

��

2

1

�

+ C1

�2

#= �i18�4t

1

(4�)2

2

�+ �nite

!;

(7.83)

where we have used the dimensional regularization to regularize the integral and p is the total incoming

four momentum.

We can determine the Higgs four-point coupling renormalization constant Z� from the counterterm

condition as follows

� i�� = �i(Z� � 1) 32� = �(iM4;scalar�loop + iM4;top�loop)

= � 9i�2

2(4�)2

�2

�+ �nite

�+ i18�4t

1

(4�)2

�2

�+ �nite

�

) Z� = 1 +

�6�

(4�)2� 24�4t�(4�)2

��1

�+ �nite

�:

(7.84)

To calculating L1 in Eq. (7.74), which is needed to calculate the �-function ��, we need to calculate

the wave function renormalization constant Z� in Eq. (7.71). The diagrams that contribute to this

renormalization constant are the Higgs propagator corrections shown in Figure 7.6.

p� q

p

q q q q

p

Figure 7.6: Loop diagrams of the Higgs propagator where the �rst diagram gives an one loop correctionsto the coupling �.

Amplitude of the �rst diagram with a top-loop is

iM2;top�loop

= (�1)3��i �tp

2

�2 �d4p

(2�)4Tr[i=pi(=p� =q)]

(p2 + i�)((p� q)2 + i�)= �6�2t

�d4p

(2�)4p(p� q)

(p2 + i�)((p� q)2 + i�)

= �6�2t�

d4p

(2�)4

� 1

0

dx1dx2p(p� q)

(x1(p2 + i�) + x2(p2 + q2 � 2pq + i�))2�(x1 + x2 � 1)

= �6�2t�

d4p

(2�)4

� 1

0

dx2p(p� q)

(p2 + i�+ x2(q2 � 2pq))2= �6�2t

�d4p

(2�)4

� 1

0

dxp2 � pq(l2 ��)2

(7.85)

Page 130 of 193


= �6�2t� 1

0

dx

�ddl

(2�)4l2 + (x2 � x)q2

(l2 ��)2= �6�2t

� 1

0

dx

�� i

(4�)d=2d

2

�(1� d=2)�(2)

�1

�

�1�d=2

+ (x2 � x)q2 i

(4�)d=2�(2� d=2)

�(2)

�1

�

�2�d=2 �;


l � p� x2q ) l2 = p2 + x22q2 � 2pqx2

� � (x2 � x)q2 � i�(7.86)

We can expand the gamma function �(x) and the function (1=�)x around �1 as follows

�(1� d=2) ' ��2

�� + 2

�and

1

(4�)d=2

�1

�

�1�d=2=

�4�

�

�2�d=21

(4�)2

�1

�

��1' 1

(4�)2�h1� (2� d=2) log� +

�

2log(4�)

i;

(7.87)

which gives us following approximation

�(1� d=2)(4�)d=2

�1

�

�1�d=2' � �

(4�)2

�2

�� log� + log(4�) + 2

�: (7.88)

By using this approximation we can write the amplitude of the top-loop diagram as

iM2;top�loop

' �6�2t� 1

0

dx

�i

(4�)22�

�2

�� log� + log(4�)

�+

i

(4�)2�

�2

�� log�� + log(4�)

��

= �3 � 6�2t� 1

0

dx(x2 � x)q2 i

(4�)2

�2

�+ : : :

�= �18�2t

��1

6

�i

(4�)2

�2

�+ : : :

�q2

= 6�2ti

(4�)21

�q2 + �nite

(7.89)

The amplitude of the other diagram with a scalar loop can be written as

iM2;scalar�loop

= (�i 32�)1

2

�d4p

(2�)4i

p2 �m2H + i�

= (�i 32�)1

2

�d4l

(2�)4i

l2 ��

= 34�

� i

(4�)d=2�(1� d=2)

�(1)

�1

�

�1�d=2!' 3

4�i

(4�)2�

�2

�� log� + log(4�) + 2

�

= 34�

i

(4�)2m2H

�2

�+ �nite

�(7.90)

We can determine the wave function renormalization constant from the counterterm condition as follows

i(q2�Z � �m) = i�q2(Z� � 1)�m2

h0Z� �m2H

�= �(iM2;top�loop + iM2;scalar�loop)

= �6�2ti

(4�)2

�1

�+ �nite

�q2 � 3

4�i

(4�)2m2H

�2

�+ �nite

�

) Z� = 1� 6�2t

(4�)2

�1

�+ �nite

�;

(7.91)

where the diagram with the scalar loop does not contribute to the renormalization constant.

Page 131 of 193


Now, we can determine L1 by substituting the renormalization constant Z� and Z� in Eq. (7.84) and

Eq. (7.91) respectively into the �rst equation in Eq. (7.71), which gives us that

ln(Z�2� Z�) =

1Xn=1

Ln(�t; �)

�n= �2 lnZ� + lnZ� = �2 ln

�1� 6

�2t(4�)2

1

�

�+

ln

�1 +

�6�

(4�)2� 24�4t�(4�)2

�1

�

�= �2

�� 6�2t(4�)2

1

�

�+

�6�

(4�)2� 24�4t�(4�)2

�1

�:

(7.92)

From this we get

L1 =12�2t(4�)2

+6�

(4�)2� 24�4t�(4�)2

: (7.93)

By substituting L1 into the �rst equation in Eq. (7.74), we get

��(�t; �) =�

�1

2�t

@

@�t+ �

@

@�

�L1 =

12

(4�)2��2t +

6

(4�)2�2 � 48

(4�)2�4t +

24

(4�)2�4t

=1

(4�)2�6�2 � 24�4t + 12��2t

�;

(7.94)

which is the �-function of the Higgs four-point coupling � in the Standard Model to one loop order:

��(�t; �) � �@�

@�=

1

(4�)2�6�2 � 24�4t + 12��2t

�: (7.95)

Page 132 of 193


Appendix F: The Scalar Sector and Vector Bosons in MWT

In this appendix, we derive the various spinor bilinears, the scalar M matrix and the vector A� matrix

in MWT theory in Chapter 3.

Spinor Bilinears

According to Ref. [10] we have that

5 =

0@ �� 0

0 � _�_�

1A ; C =

0@ �"�� 0

0 �" _� _�

1A and � =

0@ 0 ��

� _�

�� _�� 0

1A ; (7.96)

and the notation for the spinors and its adjoint are U = (UL;�; Uy _�R )T and U = (U�R; U

yL; _�)

T , respectively.

Therefore, we have that the spinor bilinears are

�UU =Uy�R UL;� + UyL; _�U_�R

�DD =Dy�R DL;� +DyL; _�D_�R

�UD =Uy�R DL;� + UyL; _�D_�R

�DU =Dy�R UL;� +DyL; _�U_�R

�U 5U =�Uy�R ; UyL; _�

�0@ �� 0

0 � _�_�

1A0@ UL�

U_�R

1A =

�� Uy�R ; UyL _�

�0@ UL�

U_�R

1A

=� Uy�R UL� + UyL _�U

_�R

�D 5D =�Dy�R DL� +DyL _�D

_�R

�D 5U =�Dy�R UL� +DyL _�U

_�R

�U 5D =� Uy�R DL� + UyL _�D

_�R

UTCU =�UL;�; U

_�R

�0@ �"�� 0

0 �" _� _�

1A0@ UL�

U_�R

1A =

�UL;�; U

_�R

�0@ �U�L�UR; _�

1A

=� UL;�U�L � U _�RUR; _�

DTCD =�DL;�D�L �D _�

RDR; _�

UTCD =� UL;�D�L � U _�

RDR; _�

DTCU =�DL;�U�L �D _�

RUR; _�

UTC 5U =�UL;�; U

_�R

�0@ �"�� 0

0 �" _� _�

1A0@ �� 0

0 �_�_

1A0@ UL;

U _ R

1A =

�UL;�; U

_�R

�0@ U�L

�UR; _�

1A

=UL;�U�L � U _�

RUR; _�

DTC 5D =DL;�D�L �D _�

RDR; _�

UTC 5D =UL;�D�L � U _�

RDR; _�

Page 133 of 193


DTC 5U =DL;�U�L �D _�

RUR; _�

�U �U =�Uy�R ; UyL; _�

�0@ 0 �� _�

�� _�� 0

1A0@ UL�

U_�R

1A = Uy�R ��

� _�U

_�R + UyL; _��

� _��UL;�

=Uy�R �� _�U

_�R + Uy

_�L ��

� _�U�L = Uy�R ��

� _�U

_�R � U�L�� _�

Uy_�

L

�D �D =Dy�R �� _�D

_�R �D�

L��

� _�Dy

_�L

�U �D =Uy�R �� _�D

_�R �D�

L��

� _�Uy

_�L

�D �U =Dy�R �� _�U

_�R � U�L�� _�

Dy_�

L

�U � 5U =�Uy�R ; UyL; _�

�0@ 0 �� _�

�� _�� 0

1A0@ �� 0

0 �_�_

1A0@ UL;

U _ R

1A = Uy�R ��

� _�U

_�R � UyL; _�� _��UL;�

=Uy�R �� _�U

_�R + U�L�

�

� _�Uy

_�L

�D � 5D =Dy�R �� _�D

_�R +D�

L��

� _�Dy

_�L

�U � 5D =Uy�R �� _�D

_�R +D�

L��

� _�Uy

_�L

�D � 5U =Dy�R �� _�U

_�R + U�L�

�

� _�Dy

_�L

UTC �U =�UL; ; U

_ R

�0@ �"� 0

0 �" _� _

1A0@ 0 ��

� _�

�� _�� 0

1A0@ UL;�

U_�R

1A

=U�L��

� _�U

_�R + UR; _��

� _��UL;� = U�L��

� _�U

_�R �D�

L��

� _�U

_�R

DTC �D =D�L�

�

� _�D

_�R �D�

L��

� _�D

_�R

UTC �D =U�L��

� _�D

_�R �D�

L��

� _�U

_�R

DTC �U =D�L�

�

� _�U

_�R � U�L�� _�

D_�R

DTC � 5D =D�L�

�

� _�D

_�R +D�

L��

� _�D

_�R

UTC � 5D =U�L��

� _�D

_�R +D�

L��

� _�U

_�R

DTC � 5U =D�L�

�

� _�U

_�R + U�L�

�

� _�D

_�R

UTC � 5U =�UL;�; U

_�R

�0@ �"�� 0

0 �" _� _�

1A0@ 0 ��

� _�

�� _�� 0

1A0@ �� 0

0 �_�_

1A0@ UL;

U _ R

1A

=U�L��

� _�U

_�R � UR; _�� _��UL;� = U�L�

�

� _�U

_�R + U�L�

�

� _�U

_�R = 2U�L�

�

� _�U

_�R

DTC � 5D =D�L�

�

� _�D

_�R +D�

L��

� _�D

_�R

UTC � 5D =U�L��

� _�D

_�R +D�

L��

� _�U

_�R


�

� _�U

_�R + U�L�

�

� _�D

_�R

Page 134 of 193


The Scalar Sector:

Now, we will derive the scalar M matrix. The charge eigenstates are

v +H �� UU + �DD; � � i( �U 5U + �D 5D)

A0 �~�3 � �UU � �DD; �0 � �3 � i( �U 5U � �D 5D)

A+ �~�1 � i~�2

p2

� �DU; �+ � �1 � i�2

p2

� i �D 5U

A� �~�1 + i~�2

p2

� �UD; �� 1 + i�2

p2

� i �U 5D

(7.97)

for the technimesons, and

�UU ��4 + i�5 +�6 + i�7

2� UTCU

�DD ��4 + i�5 ��6 � i�7

2� DTCD

�UD ��8 + i�9

p2

� UTCD

~�UU �~�4 + i~�5 + ~�6 + i~�7

2� iUTC 5U

~�DD �~�4 + i~�5 � ~�6 � i~�7

2� iDTC 5D

~�UD �~�8 + i~�9

p2

� iUTC 5D

(7.98)

for the technibaryons.

The various elements of the M matrix in terms of the bilinears in Appendix F are

M11 =U�LUL;� =

1

2

hi�4 + ~�4 + i(i�5 + ~�5) + i�6 + ~�6 + i(i�7 + ~�7)

iM21 =M12 = U�LDL;� = D�

LUL;� =1

2

hi�8 + ~�8 + i(i�9 + ~�9)

iM31 =M13 = (UyR)

�UL;� = U�L (UyR)� =

1

2

h� + i�+ i�3 + ~�3

iM41 =M14 = (DyR)

�UL;� = U�L (DyR)� =

1

2

hi�1 + ~�1 � i(i�2 + ~�2)

iM22 =D

�LDL;� =

1

2

hi�4 + ~�4 + i(i�5 + ~�5)� i�6 � ~�6 � i(i�7 + ~�7)

iM32 =M23 = (UyR)

�DL;� = D�L(U

yR)� =

1

2

hi�1 + ~�1 + i(i�2 + ~�2)

iM42 =M24 = (DyR)

�DL;� = D�L(D

yR)� =

1

2

h� + i�� i�3 � ~�3

iM33 =(UyR)

�(UyR)� =1

2

hi�4 + ~�4 � i(i�5 + ~�5) + i�6 + ~�6 � i(i�7 + ~�7)

iM43 =M34 = (DyR)

�(UyR)� = (UyR)�(DyR)� =

1

2

hi�8 + ~�8 � i(i�9 + ~�9)

iM44 =(DyR)

�(DyR)� =1

2

hi�4 + ~�4 � i(i�5 + ~�5)� i�6 � ~�6 + i(i�7 + ~�7)

i;

(7.99)

Page 135 of 193


where we have used that

�UU + �DD =Uy�R UL;� + UyL; _�U_�R +Dy�R DL;� +DyL; _�D

_�R =M31 +My

31 +M42 +My42 = 2� � �

�UU � �DD =Uy�R UL;� + UyL; _�U_�R �Dy�R DL;� �DyL; _�D _�

R =M31 +My31 �M42 �My

42 = 2~�3 � ~�3 � A0

�DU =Dy�R UL;� +DyL; _�U_�R =M41 +My

32 =~�1 � i~�2 �

~�1 � i~�2

p2

� A+

�UD =Uy�R DL;� + UyL; _�D_�R =M32 +My

41 =~�1 + i~�2 �

~�1 + i~�2

p2

� A�

i( �U 5U + �D 5D) =� iUy�R UL;� + iUyL; _� � iDy�R DL;� + iDyL; _�D_�R = �iM31 + iMy

31 � iM42 + iMy42 = 2� � �

i( �U 5U � �D 5D) =i(�Uy�R UL;� + UyL; _�U_�R +Dy�R DL;� �DyL; _�D _�

R) = i(�M31 +My31 +M42 �My

42) = 2�3 � �3 � �0

i �D 5U =i(�Dy�R UL;� +DyL; _�U_�R) = i(�M41 +My

32) = �1 � i�2 � �1 � i�2

p2

� �+

i �U 5D =i(�Uy�R DL;� + UyL; _�D_�R) = i(�M32 +My

41) = �1 + i�2 � �1 + i�2

p2

� ��

UTCU =� UL;�U�L � U _�RUR; _� =M11 �My

33 = i�4 ��5 + i�6 ��7 � �4 + i�5 +�6 + i�7

2� �UU

DTCD =�DL;�D�L �D _�

RDR; _� =M22 �My44 = i�4 ��5 � i�6 +�7 � �4 + i�5 ��6 � i�7

2� �DD

UTCD =� UL;�D�L � U _�

RDR; _� =M21 �My43 = i�8 ��9 � �8 + i�9

p2

� �UD

iUTC 5U =i[UL;�U�L � U _�

RUR; _�] = �i[M11 +My33] = �i(~�4 + i~�5 + ~�6 + i~�7) �

~�4 + i~�5 + ~�6 + i~�7

2� ~�UU

iDTC 5D =i[DL;�D�L �D _�

RDR; _�] = �i[M22 +My44] = �i(~�4 + i~�5 � ~�6 � i~�7) �

~�4 + i~�5 � ~�6 � i~�7

2� ~�DD

iUTC 5D =i[UL;�D�L � U _�

RDR; _�] = �i(M21 +My34) = �i(~�8 + i~�9) �

~�8 + i~�9

p2

� ~�UD:

Therefore, we can write the scalar charge eigenstates as follows

� = 12 (

�UU + �DD); � = i2 (

�U 5U + �D 5D);

A0 = 12 (

�UU � �DD); �0 = i2 (

�U 5U � �D 5D);

A+ = 1p2�DU; �+ = ip

2�D 5U;

A� = 1p2�UD; �� = ip

2�U 5D

(7.100)

�UU = � i2U

TCU; ~�UU = � 12U

TC 5U;

�DD = � i2D

TCD; ~�DD = � 12D

TC 5D;

�UD = � ip2UTCD; ~�UD = � 1p

2UTC 5D

(7.101)

The M matrix can be written in the form

M = Q�Q�"�� =

0BBBBBB@

U�LUL;� U�LDL;� U�LU�R;� U�LD

�R;�

D�LUL;� D�

LDL;� D�LU

�R;� D�

LD�R;�

U��R UL;� U��R DL;� U��R U�R;� U��R D�R;�

D��R UL;� D��R DL;� D��R U�R;� D��R D�R;�

1CCCCCCA: (7.102)

Page 136 of 193


There, we have that

M =

0BBBBBB@

i�UU + ~�UUi�UD+~�UDp

2�+i�+i�0+A0

2i�++A+p

2

i�UD+~�UDp2

i�DD + ~�DDi��+A�p

2�+i��i�0�A0

2

�+i�+i�0+A0

2i��+A�p

2i�UU + ~�UU

i�UD+~�UDp2

i�++A+p2

�+i��i�0�A0

2i�UD+

~�UDp2

i�DD + ~�DD

1CCCCCCA; (7.103)

because

i�UU + ~�UU =1

2

�UTCU � UTC 5U� = �UL;�U�L = U�LUL;� =M11

i�UD + ~�UDp2

=1

2

�UTCD � UTC 5D� = �UL;�D�

L = U�LDL;� = D�LUL;� =M12 =M21

� + i�+ i�0 +A0

2=1

2

�1

2( �UU + �DD � �U 5U � �D 5D)� 1

2( �U 5U � �D 5D � �UU + �DD)

�

=U�LUyR;� = Uy�R UL;� =M13 =M31

i�+ +A+

p2

=1

2

�� D 5U + �DU�= U�LD

yR;� = Dy�R UL;� =M14 =M41

i�DD + ~�DD =1

2

�DTCD �DTC 5D

�= D�

LDL;� =M22

i�� +A�p2

=1

2

�� U 5D + �UD�= D�

LUyR;� = Uy�R DL;� =M23 =M32

� + i�� i�0 �A0

2=1

2

�1

2( �UU + �DD � �U 5U � �D 5D) +

1

2( �U 5U � �D 5D � �UU + �DD)

�

=D�LD

yR;� = Dy�R DL;� =M24 =M42

i�UU + ~�UU =Uy�R UyR;� =M33

i�UD + ~�UDp2

=Uy�R DyR;� = Dy�R UyL;� =M34 =M43

i�DD + ~�DD =Dy�R DyL;� =M44:

(7.104)

According to Ref. [10] we have when we take the charge conjugated of a spinor, then we make the

substitution 0@ UL;�

U _�R

1A!

0@ UyR;�

U _�L

1A : (7.105)

The Vector Bosons:

We will now derive the vector A� matrix. The relations between the charge eigenstates and the wave-

functions of the composite objects are

v0� �A3� � �U �U � �D �D; a0� � A9� � �U � 5U � �D � 5D;

v+� �A1� � iA2�

p2

� �D �U; a+� � A7� � iA8�

p2

� �D � 5U;

v�� A1� + iA2�

p2

� �U �D; a�� A7� + iA8�

p2

� �U � 5D;

v4� �A4� � �U �U + �D �D;

(7.106)

Page 137 of 193


for the vector mesons, and

x�UU �A10� + iA11� +A12� + iA13�

2� UTC � 5U

x�DD �A10� + iA11� �A12� � iA13�

2� DTC � 5D

x�UD �A14� + iA15�

p2

� DTC � 5U

s�UD �A6� � iA5�

p2

� UTC �D

(7.107)

for the vector baryons. We can write the various elements of the A� with the bilinears in Appendix F

A�11 =U�L��

� _�U�

_�L � 1

4Q�k�

�

� _�Q�

_�;k = 12p2(A3� +A4� +A9�)

A�21 =D�L�

�

� _�U�

_�L = 1

2p2(A1� � iA2� +A7� � iA8�)

A�31 =U��R �� _�U�

_�L = 1

2p2(A10� + iA11� +A12� + iA13�)

A�41 =D��R �� _�U�

_�L = 1

2p2(�iA5� +A6� +A14� + iA15�)

A�12 =U�L��

� _�D�

_�L = 1

2p2(A1� + iA2� +A7� + iA8�)

A�22 =D�L�

�

� _�D�

_�L � 1

4Q�k�

�

� _�Q�

_�;k = 12p2(�A3� +A4� �A9�)

A�32 =U��R �� _�D�

_�L = 1

2p2(iA5� �A6� +A14� + iA15�)

A�42 =D��R �� _�D�

_�L = 1

2p2(�iA5� +A6� +A14� + iA15�)

A�13 =U�L��

� _�U

_�R = 1

2p2(A10� � iA11� +A12� � iA13�)

A�23 =D�L�

�

� _�U

_�R = 1

2p2(�iA5� �A6� +A14� � iA15�)

A�33 =U��R �� _�U

_�R � 1

4Q�k�

�

� _�Q�

_�;k = 12p2(�A3� �A4� +A9�)

A�43 =D��R �� _�U

_�R = 1

2p2(�A1� � iA2� +A7� + iA8�)

A�14 =U�L��

� _�D

_�R = 1

2p2(iA5� +A6� +A14� � iA15�)

A�24 =D�L�

�

� _�D

_�R = 1

2p2(A10� � iA11� �A12� + iA13�)

A�34 =U��R �� _�D

_�R = 1

2p2(�A1� + iA2� +A7� � iA8�)

A�44 =U��R �� _�D

_�R � 1

4Q�k�

�

� _�Q�

_�;k = 12p2(A3� �A4� �A9�);

(7.108)

which gives that the bilinears are

�U �U � �D �D =U��R �� _�U

_�R � U�L�� _�

U�_�

L �D��R �� _�D

_�R +D�

L��

� _�D�

_�L

=A�33 �A�11 �A�44 +A�22 = �p2A3� � A3� � v0�

�D �U =D��R �� _�U

_�R � U�L�� _�

D�_�

L = A�34 �A�21 = � 1p2(A1� � iA2�)

� 1p2(A1� � iA2�) � v+�

Page 138 of 193


�U �D =U��R �� _�D

_�R �D�

L��

� _�U�

_�L = A�43 �A�12 = � 1p

2(A1� + iA2�)

� 1p2(A1� + iA2�) � v��

�U �U + �D �D =U��R �� _�U

_�R � U�L�� _�

U�_�

L +D��R �� _�D

_�R �D�

L��

� _�D�

_�L

=A�33 �A�11 +A�44 �A�22 = �p2A4� � A4� � v4�

�U � 5U � �D � 5D =U��R �� _�U

_�R + U�L�

�

� _�U�

_�L �D��R ��

� _�D

_�R �D�

L��

� _�D�

_�L

=A�33 +A�11 �A�44 �A�22 =p2A9� � A9� � a0�

�D � 5U =D��R �� _�U

_�R + U�L�

�

� _�D�

_�L = A�34 +A�21 = 1p

2(A7� � iA8�)

� 1p2(A7� � iA8�) � a+�

�U � 5D =U��R �� _�D

_�R +D�

L��

� _�U�

_�L = A�43 +A�12 = 1p

2(A7� + iA8�)

� 1p2(A7� + iA8�) � a��

UTC � 5U =U�L��

� _�U

_�R + U�L�

�

� _�U

_�R = 2A�31 = 1p

2(A10� + iA11� +A12� + iA13�)

� 12 (A

10� + iA11� +A12� + iA13�) � x�UU

DTC � 5D =2D�L�

�

� _�D

_�R = 2A�42 = 1p

2(A10� + iA11� �A12� � iA13�)

� 12 (A

10� + iA11� �A12� � iA13�) � x�DD


�

� _�U

_�R + U�L�

�

� _�D

_�R = A�32 +A�41 = 1p

2(A14� + iA15�) � x�UD

UTC �D =U�L��

� _�D

_�R �D�

L��

� _�U

_�R = A�41 �A�32 = 1p

2(A6� � iA5�) � s�UD:

(7.109)

Therefore, we obtain that the charge eigenstates are

v0� = � 1p2( �U �U � �D �D); a0� = 1p

2( �U � 5U � �D � 5D);

v+� = � �D �U; a+� = �D � 5U;

v�� = � �U �D; a�� = �U � 5D;

v4� = � 1p2( �U �U + �D �D)

(7.110)

x�UU = 1p2UTC � 5U; x�DD = 1p

2DTC � 5D;

x�UD = UTC � 5D; s�UD = UTC � 5D:(7.111)

The A� matrix can be written in the form:

A�;ji =Q�i ��

� _�Q�

_�;j � 14�jiQ

�k�

�

� _�Q�

_�;k

=

0BBBBBB@

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 U��R ��

� _�D�

_�L D��R ��

� _�U

_�R D��R ��

� _�D

_�R

1CCCCCCA� 1

4�jiQ

�k�

�

� _�Q�

_�;k:(7.112)

Page 139 of 193


Finally, we can write the A� matrix as

A� =

0BBBBBB@

a0�+v0�+v4�

2p2

a+�+v+�

2

x�UUp2

x�UD

+s�UD

2

a��+v��

2�a0��v0�+v4�

2p2

x�UD

�s�UD

2

x�DDp2

x�UUp2

x�UD

�s�UD

2a0��v0��v4�

2p2

a��v��2

x�UD

+s�UD

2

x�DDp2

a+��v+�2

�a0�+v0��v4�2p2

1CCCCCCA; (7.113)

where

a0� + v0� + v4�

2p2

= 14

��U � 5U � �D � 5D � ( �U �U � �D �D)� ( �U �U + �D �D)

�= 1

4

�U��R ��

� _�U

_�R + U�L�

�

� _�U��L � (D��R ��

� _�D

_�R +D�

L��

� _�D��L )� 2U��R ��

� _�U

_�R+

2U�L��

� _�U��L

�= U�L�

�

� _�U��L � 1

4

�U�L�

�

� _�U��L +D�

L��

� _�D��L + U��R ��

� _�U

_�R+

D��R �� _�D

_�R

�= U�L�

�

� _�U�

_�L � 1

4Q�k�

�

� _�Q�

_�;k;

a+� + v+�

2= 1

2

��D � 5U � �D �U

�= 1

2

�D��R ��

� _�U

_�R + U�L�

�

� _�D�

_�L �D��R ��

� _�U

_�R+

U�L��

� _�D�

_�L

�= U�L�

�

� _�D�

_�L ;

x�UUp2

= 12U

TC � 5U = U�L��

� _�U

_�R etc.

(7.114)

Page 140 of 193


Appendix G: Discrete transformations of spinors, Q vector and M

matrix

In this appendix we will derive the discrete transformations (parity, charge conjugation and CP transfor-

mations) of spinors, the Q vector in eg. (3.7) and the M matrix in Eq. (7.102). We have that a Dirac

spinor transforms under parity as follows (acc. Eq. (40.15) in Ref. [10])

P�1U(x)P =P�1

0@ UL;�(x)

U _�R(x)

1AP = i�U(Px) = i

0@ 0 �

_�_�

�� 0

1A0@ UL;�(Px)

U _�R(Px)

1A

=

0@ iU

_�R(Px)

iUL;�(Px)

1A :

(7.115)

Therefore we have that the left- and right-handed Weyl spinors transform under parity as follows

P�1UL;�(x)P =iU _�R(Px)

P�1U _�R(x)P =iUL;�(Px);

(7.116)

and if we take the complex conjugated of them then we get

P�1Uy _�L (x)P =� iUyR;�(Px)P�1UyR;�(x)P =iUy _�L (Px):

(7.117)

We can use these transformations to write a parity transformation expression of the Q vector. The Q

vector parity transforms as follows

P�1QA�P =

0BBBBBB@

P�1UL;�(x)P

P�1DL;�(x)P

P�1U�R;�(x)P

P�1D�R;�(x)P

1CCCCCCA

=

0BBBBBB@

iU _�R(Px)

iD _�R(Px)

iU� _�L (Px)iD� _�L (Px)

1CCCCCCA

= iQ�B; _�E+AB : (7.118)

The charge conjugation transformation of a Dirac spinor (acc. Eq. (40.42) in Ref. [10]) can be written as

C�1U(x)C =C�1

0@ UL;�(Px)

U _�R(Px)

1AC = C �UT = C(Uy�)T

=

0@ "�� 0

0 " _�_�

1A24� UyL; _� Uy�R

�0@ 0 � _�_�

�� 0

1A35T

=

0@ "�� 0

0 " _�_�

1A0@ Uy�R

UyL; _�

1A =

0@ UyR;�

Uy _�L

1A :

(7.119)

Therefore we can write the C transformations of the left- and right-handed Weyl spinors as follows

C�1UL;�C =UyR;�

C�1U _�RC =Uy _�L ;

(7.120)

Page 141 of 193


and by taking the complex conjugation of them then we get

C�1UyL; _�C =UR; _�

C�1Uy�R C =U�L

: (7.121)

We can use these C transformations to write the C transformation expression of the Q vector which is

C�1QA�C =

0BBBBBB@

C�1UL;�C

C�1DL;�C

C�1U�R;�C

C�1D�R;�C

1CCCCCCA

=

0BBBBBB@

U�R;�

D�R;�

UL;�

DL;�

1CCCCCCA

= QB�E+AB : (7.122)

By using the P and C transformations expressions we can derive an expression for how the M matrix in

Eq. (7.102) transforms under P and C transformations. The P transformation of M is

P�1MAB(x)P =P�1Q�A(x)QB;�(x)P = P�1Q�A(x)PP�1QB;�(x)P

=� iQ�C; _�(Px)E+ACiQ

� _�D (Px)E+

BD = �Q�C; _�(Px)Q� _�D (Px)E+ACE

+BD

=Q� _�C (Px)Q�D; _�(Px)E+ACE

+BD = (EM�E)AB ;

(7.123)

and the C transformation of M is

C�1MABC =C�1Q�AQB;�C = C�1Q�ACC�1QB;�C = Q�CQD;�E

+ACE

+BD = (EME)AB : (7.124)

We can also make a CP transformation of the M matrix by combining the P and C transformations of

M from the two previous equations. Thus, the CP transformation of M is

(CP )�1MABCP =P�1(EME)ABP = (EEM�EE)AB =M�AB

(7.125)

Page 142 of 193


Appendix H: Witten Anomaly

We have that an SU(2) gauge theory is mathematically inconsistent if there are an odd number of left-

handed fermion doublets and no other representations in this theory.

The beginning point is that the fourth homotopy group of SU(2) is nontrivial because �4(SU(2)) = Z2.

This means that there is a gauge transformation U(x) in four-dimensional euclidean space, which "wraps"

around the gauge group such that it can not be continuously deformed to the identity. The meaning of

that the homotopy group is equal Z2 is that a gauge transformation that wraps twice around the SU(2)

group can be deformed to the identity.

To begin with we can write the euclidean path integral for the free gauge �eld A� without fermions

�DA� exp

�� 1

2g2

�d4x Tr(F��F

��)

�: (7.126)

In this path integral we are double counting because for every gauge �eld A�, there is a gauge transformed

gauge �eld

AU� = U�1A�U � iU�1@�U: (7.127)

Without fermions in the theory, the double counting cancels out when one calculates vacuum expectation

values. By including a single doublet of left-handed fermions, we now have the path integral

Z =

�DA�D D � exp

��d4x�

12g2Tr(F��F

��) + i � �D� ��

: (7.128)

We would like to integrate out the Dirac fermions which gives

�D D � exp

��d4x i � �D�

�= Det(i �D�): (7.129)

Here the right-hand side is the in�nite product of all eigenvalues of the operator i �D�. Now, with the

gauge group SU(2), a doublet of Dirac fermions is exactly the same as two left-handed or Weyl doublets.

Therefore the integration of the fermion in Eq. (7.129) with a single Weyl doublet would give the square

root of Det(i �D�). Thus for the single Weyl doublet, the partition function is

Z =

�DA�D D � Det(i �D�)

1=2 exp

��d4x 1

2g2Tr(F��F��)

�: (7.130)

An ambiguity arises here, because the square root has two signs. There is nothing that guarantees

that Det(i �D�)1=2 is invariant under the topologically non-trivial gauge transformation U . Actually,

Det(i �D�)1=2 is odd under U . We can show that for any gauge �eld A�, that

Det[i �D�(A�)]1=2 = �Det[i �D�(A

U� )]

1=2: (7.131)

I.e. if we vary the gauge �eld A� to AU� continuously, then we can end up with the opposite sign of the

square root. It is elaborated in more detail in Witten's own article about this SU(2) anomaly Ref. [19].

The consequence of this is that the partition function in Eq. (7.130) vanishes identically, because the

contribution of any gauge �eld A� is exactly cancelled by transformed gauge �eld AU� with opposite sign.

Page 143 of 193


This gives problems when we calculate the path integral ZX with insertion of any gauge invariant operator

X which is identically zero. The expectation values are indeterminate, because hXi = ZX=Z = 0=0.

Therefore, the theory is ill-de�ned.

Now let us consider some generalizations. If we have n left-handed fermion doublets, then the integration

would give [Det(i �D�)]n=2. If n is even, then the sign of the square root does nothing, but if n is

odd, then the theory is inconsistent as before. This persists even if we add additional gauge or Yukawa

couplings to the SU(2) gauge theory. Example the Standard Model of strong, weak and electromagnetic

interaction with the gauge group SU(3)C SU(2)W U(1)Y would be inconsistent if the number of left-

handed fermion doublets were odd. This is not the case in the SM, because there is a lepton left-handed

doublet for each quark doublet, and therefore the number of left-handed doublets is even.

Finally, we can consider other gauge groups than SU(2), we have

�4(SU(N)) = 0; N > 2;

�4(O(N)) = 0; N > 5;

�4(Sp(N)) = Z2; any N:

(7.132)

Thus the non-trivial conditions arise only for Sp(N) gauge groups. In conclusion, the Witten anomaly

arises when the number of left-handed doublets is odd, and it is a problem exclusively applying to Sp(N)

gauge groups, SU(2) group because SU(2) � Sp(1), and O(N < 6), except for O(2).

Page 144 of 193


Appendix I: The Electroweak Precision Parameters

The EW precision parameters called S and T describe how much the EW symmetry and custodial

symmetry are broken, respectively. We have the following de�nitions (Ref. [43])

S � �0W 3B(0);

T � g2

m2W

[�W 3W 3(0)��W+W�(0)];

W � g2m2W

2�00W 3W 3(0);

Y � g02m2W

2�00BB(0);

(7.133)

where �V1V2(q2) with V1V2 = fW 3B;W 3W 3;W+W�; BBg are the self-energy of the vector bosons shown

in Figure 7.7. The particles and are the particles in the SM and beyond the SM, respectively, that

run in the loop and couple to the EW gauge bosons. The derivative with respect to q2 of the self-energy

is denoted with a prime. The Peskin-Takeuchi parameters S and T are related to the new ones above via

(Ref. [43])�S

4s2W= S � Y �W; �T = T � s2W

1� s2WY; (7.134)

where � is the electromagnetic structure constant and sW is the weak mixing angle. Data from the LEP

experiments set the EW parameters to be (Eq. (10.72) in Ref. [73])

S = 0:05� 0:10;

T = 0:08� 0:12;

U = 0:02� 0:10

(7.135)

where the uncertainties are from the inputs. These parameters are in excellent agreement with the SM

values of zero. Values of these parameters di�erent from zero are due to new physics. The T parameter

is related to the � parameter, which is a measure for how much the custodial symmetry is broken. The

relation between these two parameters (cf. Eq. (10.68) in Ref. [73]) is

� =1

1� �(mZ)T' 1 + �(mZ)T: (7.136)

The value of the � parameter according to �(mZ) = 127:950� 0:017 (cf. Eq. (164) in Ref. [44]) and the

value of the T parameter in Eq. (7.137) is � = 1:0006� 0:0009.

V1 V2

V1 V2

Figure 7.7: The loop diagrams which give rise to the self-energy for the vector bosons V1 and V2.

By Fixing U = 0 (as is also done in Figure 7.8) moves S and T slightly upwards (cf. Eq. (10.73) in

Page 145 of 193


Ref. [73])

S = 0:07� 0:08;

T = 0:10� 0:07:(7.137)

Figure 2.2 is Figure 10.6 in Ref. [73]. We have that the black dot indicates the SM values S = T = 0,

while the red ellipse illustrates 1� constraints on S and T (for U = 0). The black dot that indicates the

SM values is inside 1� constraints, and therefore the SM precision is good. If one of the EW parameters

increases/decreases, it is needed to increase/decrease the other parameter.

Figure 7.8: 1� constraints on S and T (for U = 0) from various inputs combined with mZ . The blackdot indicates the SM values S = T = 0. The �gure is Figure 10.6 in Ref. [73].

Page 146 of 193


Appendix J: Dynkin Diagrams

In this appendix, we will use the Dynkin diagrams, as shown in Figure 7, to decompose irreducible

representations of a group into one of its subgroups.

Figure 7.9: Dynkin diagrams for simple Lie algebra (�lled dots represent short roots and hollow dots longroots).

Let G be some Lie group and G is its corresponding Lie algebra, which has the generators T a with

a = 1; : : : ;dim(G). The Lie algebra is structured with the structure constants as follows

[T a; T b] = fabcT c; (7.138)

We de�ne the rank of G as the maximal number of diagonalisable generators, which is the dimension of the

maximal Cartan subalgebra H � G. The generators of this subalgebra are Hi with i = 1; : : : ; l = rank(G),which satisfy

[Hi; Hj ] = 0: (7.139)

The rest generators are denoted E~�, which are eigenfunctions of the Cartan generators Hi

[Hi; E~�] = �iE~�; (7.140)

where the vector ~� are called a root. We can label irreps by a weight as follows

Hij�i = �ij�i (7.141)

We can use this to say something about how states are generated in a representation. Because we have

that

HiE~�j�i = (�i + �i)E~�j�i; (7.142)

then we have that the state E~�j�i is proportional to the state j� + �i. Therefore, we can build a

representation by starting with some highest-weight state and using E~� to get the other states in the

multiplet.

The Dynkin diagrams in Figure 7 are used to classify all the simple Lie (semi-simple) algebras. One

Page 147 of 193


can represent any simple Lie algebra by root diagram called Dynkin diagram, which lives in some l-

dimensional space.

The number of nodes is equal to the rank of the Lie algebra G. Each node corresponds to a simple

root, where a simple root is de�ned as the positive roots which cannot be written as a sum of the other

positive roots with positive coe�cients.

The Dynkin diagram describes the lengths and relative angles of the roots. Here is the simple roots

either long (represented by a �lled node) or short (represented by a hollow node). The lines in the diagram

which connect the nodes illustrate the angle between the two simple roots. A single line corresponds to

a angle between the two simple roots equals 120�, a double line corresponds to a angle 135�, and a triple

line corresponds to 150�. If there is no lines, then the angle is 90�. The relative length of the root is

illustrated such that a ratio of 1 for a single line,p2 for a double line and

p3 for triple line.

A useful method to presenting the interesting information in a Lie algebra is by using the Cartan

matrix, which is

Aij =2(�i; �j)

(�i; �j); (7.143)

where �i are the simple roots, (�i; �j) = j�ijj�j j cos(�) is inner product between the simple roots �i and

�j , and � is the angle between them.

ASp(4) =

0BB@ 2 �1

�2 2

1CCAASU(3) =

0BB@ 2 �1

�1 2

1CCA

Figure 7.10: The Dynkin diagram and its corresponding Cartan matrix for SU(4) and Sp(4).

We can take an example how to calculate the Cartan matrix for Sp(4), which is shown in right panel in

Figure 7.10. We use that there is 135� between the two nodes, when there is double line between them.

We have that the Cartan matrix elemenets are calculated as follows

A11 =A22 = 2

A12 =2(�1; �2)

(�2; �2)=

2j�1jj�2j cos(135�)j�2jj�2j cos(90�) = �2 j�1jj�2j

1p2= �2 1p

2

1p2= �1

A21 =2(�2; �1)

(�1; �1)=

2j�2jj�1j cos(135�)j�1jj�1j cos(90�) = �2 j�2jj�1j

1p2= �2

p2

1p2= �2

(7.144)

The general form of the Cartan matrices for SU(N + 1), SO(2N + 1), Sp(2N) and SO(2N) are

ASU(N+1) =

0BBBBBBBBBBBB@

2 �1 0 � � � 0 0

�1 2 �1 � � � 0 0

0 �1 2 � � � 0 0

� � � � � � � �0 0 0 � � � 2 �10 0 0 � � � �1 2

1CCCCCCCCCCCCA; (7.145)

Page 148 of 193


ASO(2N+1) =

0BBBBBBBBBBBB@

2 �1 0 � � � 0 0

�1 2 �1 � � � 0 0

0 �1 2 � � � 0 0

� � � � � � � �0 0 0 � � � 2 �20 0 0 � � � �1 2


ASp(2N) =

0BBBBBBBBBBBB@

2 �1 0 � � � 0 0

�1 2 �1 � � � 0 0

0 �1 2 � � � 0 0

� � � � � � � �0 0 0 � � � 2 �10 0 0 � � � �2 2


ASO(2N) =

0BBBBBBBBBBBBBBB@

2 �1 0 � � � 0 0 0

�1 2 �1 � � � 0 0 0

0 �1 2 � � � 0 0 0

� � � � � � � � �0 0 0 � � � 2 �1 �10 0 0 � � � �1 2 0

0 0 0 � � � �1 0 2

1CCCCCCCCCCCCCCCA

: (7.148)

According to Dynkin the rank(G) simple roots of the Lie algebra are given by the rows of the Cartan

matrix. Therefore, to build representations we start with some highest weight state �. To construct the

other weight states of the irrep we act with E�~�, which takes the state j�i to the state j�� i.We can take an example with the 3-dimensional irrep of SU(3), where the simple roots are �1 = (2;�1)

and �2 = (�1; 2) according to the Cartan matrix of SU(3) in left panel in Figure 7.10. We choose to

start with the highest weight state w1 = (1; 0). The procedure is to subtract the simple roots. We get

the weight w2 = (�1; 1) when we subtract �1, and w3 = (0;�1) when we subtract �2 from w2. Since

there are no more positive components, then we stop. This 3-dimensional irrep f(1; 0); (�1; 1); (0;�1)gis known as the fundamental representation of SU(3). In the following we will give some useful examples,

where we decompose the irrep of a group under a subgroup.

Decomposition of the 5-dimensional irrep of Sp(4) into the subgroup SO(4)

We will give an example here, where we decompose the 5-dimensional irreducible representation of Sp(4)

into a (2; 2)+(1; 1) of SO(4). Firstly, we need the Cartan matrix of Sp(4), which according to Eq. (7.147)

is

ASp(4) =

0@ 2 �1�2 2

1A ; (7.149)

Page 149 of 193


so we have the simple roots in the Dynkin basis are given by

�1 = (2;�1) and �2 = (�2; 2): (7.150)

We can look at the irreducible representations which are generated by the highest weight state w1 = (0; 1)

with the dimension 5. We can �rst subtract �2 to get w2 = (2;�1). We can now subtract �1 from this

to get w3 = (0; 0). Subtracting �1 from this gives w4 = (�2; 1). Finally, subtracting �2 from w4

gives w5 = (0;�1), which is where we stop. Putting these together we get a 5-dimensional irreducible

representation of Sp(4). Now, we wish to �nd out how does this irreducible representation decompose

under SO(4) � SU(2) SU(2) subgroup.

We split the roots (a1; a2) into ((a1); (a2)), where the elements tell us which SU(2) we have. We can

decompose the roots w1;2;:::;5 which make up the 5-dimensional irreducible representation as

w1 = ((0); (1)); w2 = ((2); (�1)); w3 = ((0); (0));

w4 = ((�2); (1)); and w5 = ((0); (�1));(7.151)

The 2-dimensional irreducible representation of SU(2) have the weight states 1 and �1. The 3-dimensional

irrep of SU(2) have the weight states 2, 0 and �2. The 4-dimensional irrep of SU(2) have the weight

states 3, 1, �1 and �3, and so on for higher dimensional irrep.

We can identify a triplet in the left SU(2) group between the weights w2, w4 and w5 and a singlet

w1. This make up two SU(2) doublets, because we have 2 2 = 1� 3. We can also identify two doublets

under the right SU(2) group in these four weights. The last weight w3 is both a singlet under both

SU(2) groups. From this, we can see that fw1; w2; w4; w5g corresponds to a (2; 2) of SO(4), and fw3gcorresponds to a (1; 1) of SO(4). Thus, a 5-dimensional irrep of Sp(4) decomposes into a (2; 2)+ (1; 1) of

the subgroup SO(4) 2 Sp(4).

Decomposition of the 10-dimensional irrep of Sp(4) into the subgroup SO(4)

We will give an example here, where we decompose the 10-dimensional irreducible representation of Sp(4)

into a (3; 1) + (1; 3) + (2; 2) of SO(4). Firstly, we need the Cartan matrix of Sp(4), which according to

Eq. (7.147) is

ASp(4) =

0@ 2 �1�2 2

1A ; (7.152)

where the simple roots in the Dynkin basis are given by

�1 = (2;�1) and �2 = (�2; 2): (7.153)

We can look at the irrep which are generated by highest weight state w1 = (2; 0). We can generate these

irrep by subtracting the simple roots in the Dynkin basis which are �1 = (2;�1) and �2 = (�2; 2). We

get the weight states

Page 150 of 193


w1 = ((2); (0)); w2 = ((0); (1)); w3 = ((�2); (2));w4 = ((2); (�1)); w5 = ((0); (0)); w6 = ((0); (0));

w7 = ((�2); (1)); w8 = ((2); (�2)); w9 = ((0); (�1));w10 = ((�2); (0));

(7.154)

We can identify a triplet under the left SU(2) group between the weights w1, w5 and w10, and a triplet

under the right SU(2) group between w3, w6 and w8. We can also identify a triplet under the left SU(2)

group between the weights w4, w7 and w9 and a singlet w2. This make up two SU(2) doublets, because

we have 2 2 = 1 � 3. In the same weights, we can identify two doublets under the right SU(2) group.

From this, we can see that fw1; w5; w10g corresponds to a (3; 1), fw3; w6; w8g corresponds to a (1; 3), and

fw2; w4; w7; w9g corresponds to a (2; 2) of SO(4). Thus, a 10-dimensional irrep of Sp(4) decomposes into

a (3; 1) + (1; 3) + (2; 2) of the subgroup SO(4) 2 Sp(4).

Page 151 of 193


Appendix K: Nonlinear Realization

We will examine the nonlinear realization of SU(4)=Sp(4) in this appendix. The Nambu-Goldstone bosons

can be realized as the broken symmetry excitations of the vacuum � in Eq. (7.251) in Appendix C-4 as

follows

�! ��T ; (7.155)

where

� = exp(i�); (7.156)

where � is a linear combination of the broken generators Xa de�ned in Appendix A. For Sp(4) transfor-

mations V 2 Sp(4) the condensate is invariant, i.e.

� = V �V T : (7.157)

We de�ne that the nonlinear realization of SU(4)=Sp(4) transforms under U 2 SU(4) as follows

��T ! U��TUT ; (7.158)

and therefore we have that the exponential realization transforms as

� ! U�V y: (7.159)

We de�ne the semi-covariant derivative

D�� = @�� iA��; (7.160)

where A� are gauge �elds for SU(4). The quantity

C� = i�yD�� Ka�T

a (7.161)

transforms like a Sp(4) gauge �eld as follows

C� = i�y(@� � iA�)� !iV �yV y(@� � iVA�V y + V @�V

y)V �V y = iV �y(@��)V y + iV @�Vy + V �yA��V y =

V (i�y(@� � iA�)�)V y + V i@�Vy = V (C� + i@�)V

y;

(7.162)

where we have used that @�(V Vy) = 0) (@�V )V

y = �V @�V y and A� ! VA�V y � V @�V y.We can project C� onto �elds parallel and perpendicular to the unbroken Sp(4) direction,

C?� = 2Tr(C�Xa)Xa = 2Tr(Kb

�TbXa)Xa = �abKb

�Xa = Ka

�Xa

Ck� = 2Tr(C�Si)Si = 2Tr(Kb

�TbSi)Si = �ibKb

�Si = Ki

�Si:

(7.163)

We can rewrite these as follows

Page 152 of 193


C?� =K�X =1

2K�(S +X � S +X) =

1

2K�

�S +X +�(ST +XT )�y

�=1

2

�K�(S +X) + �K�(S

T +XT )�y�=

1

2(C� +�CT��

y)

Ck� =K�S =1

2K�(S +X + S �X) =

1

2K�

�S +X � �(ST +XT )�y

�1

2

�K�(S +X)� �K�(S

T +XT )�y�=

1

2(C� � �CT��

y);

(7.164)

where the relations S� + �ST = 0 and X� � �XT = 0 are been used, and it is been de�ned that

C� = C?� + Ck�. These transform as follows

C?� =1

2(C� +�CT��

y)!1

2

�V (C� + i@�)V

y + V �V T (V (C� + i@�)Vy)TV ��yV y

�=

1

2V�C� + i@� +�CT��

y + i�V T (@�V�)�y

�V y = V C?� V

y

(7.165)

and

Ck� =1

2(C� � �CT��

y)!1

2

�V (C� + i@�)V

y � V �V T (V (C� + i@�)Vy)TV ��yV y

�=

1

2V�C� + i@� � �CT��

y � i�V T (@�V �)�y�V y = V (Ck� + i@�)V

y;

(7.166)

where we have used that �V T (@�V�)�y = V y(@�V ) = �(@�V y)V , because �V T = V yV �V T = V y�

and �V � = V �V TV � = V �.

Page 153 of 193


Appendix C-2: Introduction to Elementary Particle Physics

Standard Model

In the future, we de�ne i to be a Dirac spinor in the generation i, which either can be ui, di, �i or ei.

The left- and the right-handed can be projected out with the projection operators PL;R = (1� 5)=2 as

follows L;R = PL;R . So far all the fermions are massless. A Dirac mass term is not allowed, because

the SU(2)L symmetry transforms the �eld eL into another �eld �L. Under such a transformation the

mass term

�m � =�m � 12 � + 12� 5 5

�= �m

�12� � 1

2 y y5 0 5

�

=�m y

1� y52

01 + 5

2 + y

1 + y52

01� 5

2

!

=�m( � L R + � R L):

(7.167)

is clearly not invariant, and therefore it is forbidden.

The �rst term in the Yukawa Lagrangian in Eq. (2.26) can be written as

� �Q0LI GuIJu

0RJ �c =� �u0LI G

uIJu

0RJ �

0� + �d0LI GuIJu

0RJ �

�

=� �uLLUu;LLI G

uIJU

u;RyJL uRL�

0� + �dLNUd;LNI G

uIJU

u;RyJL uRL�

�

=� �uLLUu;LLI G

uIJU

u;RyJL uRL�

0� + �dLNUd;LNMU

u;LyML Uu;LLI G

uIJU

u;RyJL uRL�

�

=�XI

�uLI

p2mu;I

vuRI �

0� +XI;J

�dLI VqyIJ

p2mu;J

vuRJ �

�

=�XI

uyI1� 5

2 0p2mu;I

v

1 + 5

2uI

1p2(v + h� i�3) +

XI;J

�dLI VqyIJ

p2mu;J

vuRJ �

�

=�XI

mu;I

2v�uI(1 + 5)uI(v + h� i�3) +

XI;J

�dLI VqyIJ

p2mu;J

vuRJ �

�;

(7.168)

and the second term is

� �Q0LI GdIJd

0RJ � =� �u0LI G

dIJd

0RJ �

+ � �d0LI GuIJd

0RJ �

0

=� �uLLUu;LLI G

dIJU

d;RyJL dRL�

+ � �dLNUd;LNI G

dIJU

d;RyJL dRL�

0

=� �uLNUu;LNMU

d;LyML U

d;LLI G

dIJU

d;RyJL dRL�

+ � �dLLUd;LLI G

dIJU

d;RyJL dRL�

0

=�XI;J

�uLI VqIJ

p2mu;J

vdRJ �

+ �Xi

�dLI

p2md;I

vdRI �

0

=�XI;J

�uLI VqIJ

p2mu;J

vdRJ �

+ �XI

dyI1� 5

2 0p2md;I

v

1 + 5

2dI

1p2(v + h� i�3)

=�XI;J

�uLI VqIJ

p2mu;J

vdRJ �

+ +XI

md;I

2v�dI(1 + 5)dI(v + h+ i�3);

(7.169)

and the third term is

Page 154 of 193


��L0LI GeIJe

0RJ � =� ��0LI G

eIJe

0RJ �

+ � �e0LI GeIJe

0RJ �

0

=� ��LLU�;LLI G

eIJU

e;RyJL eRL�

+ � �eLNUe;LNI G

eIJU

e;RyJL eRL�

0

=� ��LNU�;LNMU

e;LyML U

e;LLI G

eIJU

e;RyJL eRL�

+ � �eLLUe;LLI G

eIJU

e;RyJL eRL�

0

=�XI;J

��LI VlIJ

p2me;J

veRJ �

+ �Xi

�eLI

p2me;I

veRI �

0

=�XI;J

��LI VlIJ

p2me;J

veRJ �

+ �XI

eyI1� 5

2 0p2me;I

v

1 + 5

2eI

1p2(v + h� i�3)

=�XI;J

��LI VlIJ

p2me;J

veRJ �

+ +XI

me;I

2v�eI(1 + 5)eI(v + h+ i�3);

(7.170)

where we have used Eq. (2.30), Eq. (2.31), anticommutator f 5 �g = 0, and that R;L = ((1� 5)=2) .We have that

( � )y = yL; _� _�R + y�R L;� = � ;

( � 5 )y =� yL; _� _�R + y�R L;� = � � 5

(7.171)

according to the expressions of the spinor bilinears � and � 5 in Appendix F.

Unitarity of WLWL Scattering Amplitude

We need to determine the Mandelstam variables in the scattering amplitudes, which are

s = (p1 + p2)2 = (q1 + q2)

2 = 2m2W + 2p1 � p2 = 2m2

W + 2q1 � q2;t = (p1 � q1)2 = (p2 � q2)2 = 2m2

W � 2p1 � q1 = 2m2W � 2p2 � q2;

u = (p1 � q2)2 = (p2 � q1)2 = 2m2W � 2p2 � q1 = 2m2

W � 2p1 � q2;

(7.172)

where the sum of them gives s + t + u = 4m2W . Therefore, the products between the four-momentum

vectors are

p1 � p2 = q1 � q2 = 1

2s�m2

W ;

p1 � q1 = p2 � q2 = m2W � 1

2t;

p2 � q1 = p1 � q2 = m2W � 1

2u;

p21 = p22 = q21 = q22 = m2W :

(7.173)

The sum of the gauge diagrams in Figure 2.1 is


t

=e2

4s2Wm4W

�s2 + 4st+ t2 � 4m2

W (s+ t)� 8m2W

sut� s(t� u) + 3m2

W (t� u)� t(s� u)+

3m2W (s� u)� 8m2

W

su2�+O

��EmW

�0�

=e2

4s2Wm4W

�(4m2

W � u)2 + 2st+ 4m2Wu+

8m2W

s(s+ t)u� 8m2

W

sut� st+ u(4m2

W � u)�

(7.174)

Page 155 of 193


3m2Wu� st� 6m2

Wu

�+O

��EmW

�0�

=e2

4s2Wm4W

�� 8m2

Wu+ 4m2Wu+ 8m2

Wu+ 4m2Wu� 3m2

Wu� 6m2Wu

�+O

��EmW

�0�

= � e2

4s2Wm4W

u+O��

EmW

�0�:

The Longitudinal Polarization Four-Vectors

Let us derive the longitudinal polarization four-vectors. We have that the particle moves in the energy-

momentum four-vector direction, which can be written as

k� =�k0;~k

�: (7.175)

The conditions for we have a longitudinal polarization vector is the inner product between them and the

mommentum vector is 0 and normalized to be -1, therefore we have that

��k� = 0; �� = �1: (7.176)

Thus, the longitudinal polarization vectors can be written as follows

��L(k) =1

mW

j~kj;

~k

j~kjk0

!; (7.177)

which obeys the conditions

��Lk� =j~kjk0mW

�~k � ~kmW j~kj

k0 = 0;

��L�L;� =j~kj2m2W

�~k~k

m2W j~kj2

�k0�2

=1

m2W

�j~kj2 � j~kj �m2

W

�= �1:

(7.178)

The time-like and the spatial components of the polarization vector can be rewritten as follows

j~kjm2W

=(k0 + j~kj)j~kjmW (k0 + j~kj)

=

�k0�2

+ k0j~kj �m2W

mW (k0 + j~kj)=

k0

mW� mW

k0 + j~kj;

~kk0

mW j~kj=~k

k0j~kj+ �k0�2mW (k0 + j~kj)j~kj

= ~kk0j~kj+ j~kj2 +m2

W

mW (k0 + j~kj)j~kj=

~k

mW+

mW

k0 + j~kj~k

j~kj;

(7.179)

where it is used that�k0�2

= j~kj2+m2W . Therefore the longitudinal polarization vectors can be rewritten

as

��L(k) =1

mW

j~kj;

~k

j~kjk0

!=

1

mW

�k0;~k

�+

mW

k0 + j~kj

�1;

~k

j~kj

!(7.180)

In the special case where the energy is much larger than the mass of the particle we can make the

approximation

j~kj =q(k0)

2 �m2W ' k0: (7.181)

Thus, the longitudinal polarization vector can be written as

Page 156 of 193


��L(k) =1

mW

�k0;~k

�+

mW

k0 + j~kj

�1;

~k

j~kj

!' 1

mW

�k0;~k

�� mW

2 (k0)2

�k0;�~k

�(7.182)

In the center-of-mass frame of the incoming W+(p1)W�(p2) pair, where ~p1 = �~p2, we can express

the longitudinal polarization four-vector as

��L(p1) =p�1mW

� 2mW

sp�2 (7.183)

and similarly

��L(p2) =p�2mW

� 2mW

sp�1 ; (7.184)

where s = (p1 + p2)2 = 4 (p0)

2.

Amplitudes of the WLWL Diagrams

In this subsection, we calculate the amplitudes of the diagrams for the WLWL scattering in Figure 2.1

and Figure 2.2. To this we use the longitudinal polarization four-vectors in Eq. (7.183), (7.183) and these

for the outgoing W bosons WL(q1)WL(q2). We need also to determine the Mandelstam variables, which

are

s = (p1 + p2)2 = (q1 + q2)

2 = 2m2W + 2p1 � p2 = 2m2

W + 2q1 � q2;t = (p1 � q1)2 = (p2 � q2)2 = 2m2

W � 2p1 � q1 = 2m2W � 2p2 � q2;

u = (p1 � q2)2 = (p2 � q1)2 = 2m2W � 2p2 � q1 = 2m2

W � 2p1 � q2;

(7.185)

where the sum gives s+ t+ u = 4m2W . Therefore, the products between the four-momentum vectors are

p1 � p2 = q1 � q2 = 1

2s�m2

W ;

p1 � q1 = p2 � q2 = m2W � 1

2t;

p2 � q1 = p1 � q2 = m2W � 1

2u;

p21 = p22 = q21 = q22 = m2W :

(7.186)

Let us start with the contributions from the gauge diagrams in Figure 2.1, which are being calculated

for the diagram of a four-point vertex as follows

iM4(WL;WL !WL;WL)

= ie2

s2W[2�L(p2) � �L(q1)�L(p1) � �L(q2)� �L(p2) � �L(p1)�L(q1) � �L(q2)

� �L(p2) � �L(q2)�L(p1) � �L(q1)]

' ie2

s2W

�2

�p2mW

� 2mW

sp1

��q1mW

� 2mW

sq2

��p1mW

� 2mW

sp2

��q2mW

� 2mW

sq1

��

p2mW

� 2mW

sp1

��p1mW

� 2mW

sp2

��q1mW

� 2mW

sq2

��q2mW

� 2mW

sq1

��

p2mW

� 2mW

sp1

��q2mW

� 2mW

sq1

��p1mW

� 2mW

sp2

��q1mW

� 2mW

sq2

��

Page 157 of 193


= ie2

s2W

�2

�1

m2W

p2 � q1 + 4m2W

s2p1 � q2 � 2

s(p1 � q1 + p2 � q2)

��1

m2W

p1 � q2 + 4m2W

s2p2 � q1

� 2

s(p1 � q1 + p2 � q2)

��

1

m2W

p1 � p2 + 4m2W

s2p1 � p2 � 2

s(p22 + p21)

��1

m2W

q1 � q2

+4m2

W

s2q1 � q2 � 2

s(q21 + q22)

��

1

m2W

p2 � q2 + 4m2W

s2p1 � q1 � 2

s(p2 � q1 + p1 � q2)

��

1

m2W

p1 � q1 + 4m2W

s2p2 � q2 � 2

s(p1 � q2 + p2 � q1)

��(7.187)

' ie2

s2W

�2

�1

m4W

�m2W � 1

2u�2 � 8

sm2W

�m2W � 1

2u��m2W � 1

2t�� 1

m4W

�12s�m2

W

�2� 8

sm2W

�12s�m2

W

�m2W

��

1

m4W

�m2W � 1

2t�2 � 8

sm2W

�m2W � 1

2t��m2W � 1

2u��

= ie2

s2W

�� 2

m2W

u+1

2m4W

u2 � 1

4m4W

s2 +1

m2W

s+1

m2W

t� 1

4m4W

t2 +8

sm2W

�� 1

4ut��

= ie2

4m4W s

2W

�� 8m2

Wu+ 2u2 � s2 + 4m2W s+ 4m2

W t� t2 �8m2

W

sut

�

= ie2

4m4W s

2W

�� 8m2

Wu+ 32m4W � 16m2

W s� 16m2W t+ 2s2 + 4st+ 2t2 � s2 + 4m2

W s

+ 4m2W t� t2 �

8m2W

sut

�' i

e2

4m4W s

2W

�s2 + 4st+ t2 � 4m2

W (s+ t)� 8m2W

sut

�;

where we have used

2u2 = 2(4m2W � s� t)2 = 32m4

W � 16m2W s� 16m2

W t+ 2s2 + 4st+ 2t2:

We have ignored the terms with order lower than O((E=mW )0) because these have no in�uence on the

unitarity. Thus, we have that the amplitude is

M4(WL;WL !WL;WL) =e2

4m4W s

2W

�s2 + 4st+ t2 � 4m2

W (s+ t)� 8m2W

sut

�+O

��EmW

�0�:

(7.188)

The next Feynman diagram is the diagram with , Z propagator in s channel in Figure 2.1, which

amplitude is

iMZ; s (WL;WL !WL;WL) = �ie2

�1

s+c2w=s

2W

s�m2Z

� h(p1 � p2)��L(p1) � �L(p2)

+ 2p2 � �L(p1)��L(p2)� 2p1 � �L(p2)��L(p1)ih(q2 � q1)��L(q1) � �L(q2)� 2q2 � �L(q1)�L;�(q2)

+ 2q1 � �L(q2)�L;�(q1)i

'� ie2 1s

�(p1 � p2)�

�p1mW

� 2mW

sp2

��p2mW

� 2mW

sp1

�+ 2p2 �

�p1mW

� 2mW

sp2

��p�2mW

� 2mW

sp�1

�� 2p1 �

�p2mW

� 2mW

sp1

��p�1mW

� 2mW

sp�2

��

(q2 � q1)��q1mW

� 2mW

sq2

��q2mW

� 2mW

sq1

�� 2q2 �

�q1mW

� 2mW

sq2

��q2�mW

� 2mW

sq1�

�+ 2q1 �

�q2mW

� 2mW

sq1

��q1�mW

� 2mW

sq2�

��

Page 158 of 193


=� i e2

s2W

1

s

�(p1 � p2)�

�p1 � p2m2W

� 4

sp21 +

4m2W

s2p1 � p2

�+ 2

�p1 � p2m2W

p�2

� 2

sp1 � p2p�1 �

2

sp22p

�2 +

4m2W

s2p22p

�1

�� 2

�p1 � p2m2W

p�1 �2

sp1 � p2p�2

� 2

sp21p

�1 +

4m2W

s2p21p

�2

��(q2 � q1)�

�q1 � q2m2W

� 4

sq21 +

4m2W

s2q1 � q2

�

� 2

�q1 � q2m2W

q2� � 2

sq1 � q2q1� � 2

sq22q2� +

4m2W

s2q22q1�

�+ 2

�q1 � q2m2W

q1�

� 2

sq1 � q2q2� � 2

sq21q1� +

4m2W

s2q21q2�

��

'� i e2

s2W

1

s

�p1 � p2m2W

(p�2 � p�1 ) +4

sp1 � p2(p�2 � p�1 )

��q1 � q2m2W

(q1� � q2�) + 4

sq1 � q2(q1� � q2�)

�

=� i e2

s2W

1

s

�1

m4W

+8

sm2W

+16

s2

�(p1 � p2)(q1 � q2)(p�2 � p�1 )(q1� � q2�)

'� i e2

s2W

1

s

�1

m4W

+8

sm2W

+16

s2

��1

4s2 �m2

W s+m4W

�(t� u)

=� e2

4m4W s

2W

�s(t� u)� 3m2

W (t� u)�;


�p�2 � p�1

��q1� � q2�

�= p2 � q1 � p2 � q2 � p1 � q1 + p1 � q2 = t� u:

We have again ignored the terms with order lower than O((E=mW )0). Thus, we have that the scattering

amplitude is

MZ; s (WL;WL !WL;WL) = � e2

4m4W s

2W

�s(t� u)� 3m2

W (t� u)�+O

��EmW

�0�: (7.189)

The last Feynman diagram in Figure 2.1 is the diagram with , Z propagator in t channel, which

amplitude is

iMZ; t (WL;WL !WL;WL) = �ie2

�1

s+c2w=s

2W

s�m2Z

� h(p1 + q1)

��L(p1) � �L(q1)

� 2q1 � �L(p1)��L(q1)� 2p1 � �L(q1)��L(p1)ih(p2 + q2)��L(p2) � �L(q2)� 2q2 � �L(p2)�L;�(q2)

� 2p2 � �L(q2)�L;�(p2)i

'� i e2

s2W

1

t

�(p1 + q1)

�

�p1mW

� 2mW

sp2

��q1mW

� 2mW

sq2

�� 2q1 �

�p1mW

� 2mW

sp2

��p�1mW

� 2mW

sq�2

�� 2p1 �

�q1mW

� 2mW

sq2

��p�1mW

� 2mW

sp�2

��

(p2 + q2)�

�p2mW

� 2mW

sp1

��q2mW

� 2mW

sq1

�� 2q2 �

�p2mW

� 2mW

sp1

��q2�mW

� 2mW

sq1�

�� 2p2 �

�q2mW

� 2mW

sq1

��q2�mW

� 2mW

sq1�

��

=� i e2

s2W

1

t

�� p1 � q1

m2W

(p�1 + q�1 ) +4

sp1 � q1(p�2 + q�2 ) +

4m2W

s2

�p2 � q2(p�1 + q�1 ) (7.190)

� 2p1 � q2(p�1 + q�2 )

�� p2 � q2

m2W

(p2� + q2�) +4

sp2 � q2(p1� + q1�)

Page 159 of 193


+4m2

W

s2

�(p1 � q1)(p2� + q2�)� 2p1 � q2(p1� + q1�)

��

'� i e2

s2W

1

t

�1

m4W

(p1 + q1) � (p2 + q2)� 4

m2W s

(p1 + q1) � (p1 + q1)� 4

m2W s

(p2 + q2)�

(p2 + q2) +16

s2(p2 + q2) � (p1 + q1)

�(p1 � q1)2

=� i e2

s2W

1

t

�1

m4W

(s� u) + 16

s2(s� u)� 8

m2W s

(4m2W � t)

��m4W �m2

W t�1

4t2�

'� i e2

4m4W s

2W

�t(s� u)� 3m2

W (s� u) + 8m2W

su2�;


(p1 + q1) � (p2 + q2) = p1 � p2 + p1 � q2 + q1 � p2 + q1 � q2 = s� u;(p1 + q1) � (p1 + q1) = p21 + q21 + 2p1 � q1 = 4m2

W � t;(p2 + q2) � (p2 + q2) = 4m2

W � t:

We have again ignored the terms with order lower than O((E=mW )0). Thus, we have that the scattering

amplitude is

MZ; t (WL;WL !WL;WL) = � e2

4m4W s

2W

�t(s� u)� 3m2

W (s� u) + 8m2W

su2�+O

��EmW

�0�:

(7.191)

Now, we will calculate the contributions from the Higgs boson in Figure 2.2 to the scattering amplitude.

The contribution with the Higgs boson propagator in s is

iMHiggss (WL;WL !WL;WL)

= ��L(p1)iemW

sWg��

�L(p2)

i

(p1 + p2)2 �m2h

��L(q1)iemW

sWg��

�L(q2)

' �ie2m2

W

s2W

1

m4W

(p1 � p2)(q1 � q2) 1

s�m2h

= �i e2

s2Wm2W

�1

2s�m2

W

�21

s�m2h

= �i e2

4s2Wm2W

�s� 2m2

W

�2s�m2

h

+O��

EmW

�0�:

(7.192)

The contribution with the Higgs boson propagator in t channel is

iMHiggst (WL;WL !WL;WL)

= ��L(q1)iemW

sWg��

�L(p1)

i

(p1 � q1)2 �m2h

��L(q2)iemW

sWg��

�L(p2)

' �ie2m2

W

s2W

1

m4W

(q1 � p1)(q2 � p2) 1

t�m2h

= �i e2

s2Wm2W

�m2W � 1

2t

�21

t�m2h

= �i e2

4s2Wm2W

�t� 2m2

W

�2t�m2

h

+O��

EmW

�0�:

(7.193)

The sum of the gauge diagrams in Figure 2.1 is


t =

Page 160 of 193


e2

4s2Wm4W

�s2 + 4st+ t2 � 4m2

W (s+ t)� 8m2W

sut� s(t� u) + 3m2

W (t� u)� t(s� u)+

3m2W (s� u)� 8m2

W

su2�+O

��EmW

�0�=

e2

4s2Wm4W

�(4m2

W � u)2 + 2st+ 4m2Wu+

8m2W

s(s+ t)u� 8m2

W

sut� st+ u(4m2

W � u)�

3m2Wu� st� 6m2

Wu

�+O

��EmW

�0�=

e2

4s2Wm4W

�� 8m2

Wu+ 4m2Wu+ 8m2

Wu+ 4m2Wu� 3m2

Wu� 6m2Wu

�+O

��EmW

�0�=

� e2

4s2Wm4W

u+O��

EmW

�0�:

The gauge structure ensures the cancellation of the O(E4=m4W ) terms. The problem is that the sum

of the gauge diagrams are left with O(E2=m2W ). Therefore, for the scattering amplitudes with purely

gauge bosons without Higgs bosons then the amplitudes grow with the energy as s=m2W , and thus it is

not unitarized.

However, we have the contributions from the Higgs diagrams in Figure 2.2, which are

MHiggs (WL;WL !WL;WL)

=MHiggss +MHiggs

t = � e2

4s2Wm2W

"�s� 2m2

W

�2s�m2

h

+

�t� 2m2

W

�2t�m2

h

#+O

��EmW

�0�

' � e2

4s2Wm2W

(s+ t) +O��

EmW

�0�= � e2

4s2Wm2W

�4m2

W � u�+O�� EmW

�0�

' e2

4s2Wm2W

u+O��

EmW

�0�;

(7.194)

in the limit s � m2h;m

2W . Therefore, the total amplitude consists only of terms of order larger than

O((E=mW )0), i.e.

MTotal =MHiggs +MGauge = O��

EmW

�0�: (7.195)

Higgs Mass Corrections

In this subsection, some of the longer calculations of the corrections of the mass of the Higgs boson are

represented.

To calculating the mass correction to �rst loop order, we need to calculate the sum of all one-particle-

irreducible diagrams for the Higgs propagator, which are shown in Figure 2.10. To this work, we will

�rstly calculate two useful integrals. The �rst integral is calculated in the following way

�d4p

(2�)41

p2 �m2f + i�

=

�d3p

(2�)4

�dp0

1

p20 � ~p2 �m2f + i�

=

�d3p

(2�)4

� 1

�1dp0

1

(p0 +q~p2 +m2

f )(p0 �q~p2 +m2

f ) + i�=

�d3p

(2�)4i2�

2q~p2 +m2

f

(7.196)

Page 161 of 193


=i

8�2

� �

0

dj~pj ~p2q~p2 +m2

f

=i

8�21

2

"p2

s1 +

m2f

p2�m2

f ln

p+ p

s1 +

m2f

p2

!#p=�p=0

=i

16�2

"�2

s1 +

m2f

�2�m2

f ln

� + �

q1 +

m2f

�2

mf

!#� i

16�2

��2 �m2

f ln

�2�

mf

��;

(7.197)

where we have used Cauchy's residue theorem to solve the integral, and we have made a hard cuto� at

�. In the last step we assume that �� mf . The second useful integral is solved as follows

�d4p

(2�)41

(p2 �m2f + i�)((p� q)2 �m2

f + i�)=

�d4p

(2�)41

(p2 �m2f + i�)(p2 + q2 � 2pq �m2

f + i�)

=

�d4p

(2�)4

� 1

0

dx1

(1� x)(p2 �m2f ++i�) + x(p2 + q2 � 2pq �m2

f + i�)=

�d4l

(2�)4

� 1

0

dx1�

l2 ��+ i��2

=

� 1

0

dxi

(2�)4

�d4

� �

0

dlEl3E

(lE +�)2=

� 1

0

dxi

8�2

��

�+ l2E+ ln(� + l2E)

�lE=�lE=0

� � i

8�2

� 1

0

dx

�1 + ln

��

�2

��= � i

8�2

�1 +

� 1

0

dx ln

��

�2

��; (7.198)

where we have used following de�nitions

l � p� xq ) l2 = p2 + x2q2 � 2xqp;

� � �x(1� x)q2 + xm2 + (1� x)m2:

To solving this integral we have also carried out a Wick rotation, where we make the substitutions l0E = il0

and ~lE = ~l.

We have the Higgs propagator with a fermion loop which is calculated as follows

�i�fermion�loop =� e2

s2W

m2f

4m2W

�d4p

(2�)4Tr((p=+mf )((p=� q=) +mf )

(p2 �m2f + i�)((p� q)2 �m2

f + i�)(7.199)

=� 4e2

s2W

m2f

4m2W

�d4p

(2�)4p(p� q) +m2

f

(p2 �m2f + i�)((p� q)2 �m2

f + i�)

=� 21

2

e2

s2W

m2f

4m2W

�d4p

(2�)4(p2 �m2

f ) + ((p� q)2 �m2f )� q2 + 4m2

f

(p2 �m2f )((p� q)2 �m2

f )

=� 2e2

s2W

m2f

4m2W

�d4p

(2�)4

"1

(p� q)2 �m2f

+1

p2 �m2f

+4m2

f � q2(p2 �m2

f )((p� q)2 �m2f )

#

=� 2e2

s2W

m2f

4m2W

�d4p

(2�)4

"2

1

p2 �m2f

+4m2

f � q2(p2 �m2

f )((p� q)2 �m2f )

#

�� 4e2

s2W

m2f

4m2W

i

16�2

��2 �m2

f ln

�2�

mf

��+

e2

s2W

m2f

4m2W

i

4�2(4m2

f �m2H)

� 1

0

dx

�1 + ln

��x(1� x)q2 +m2f

�2

��;

where both the �rst and the second integral in Eq. (2.101) and Eq. (2.102) are been used.

The diagrams with a Z='Z and a W�='� in Figure 2.10 give

Page 162 of 193


�i�Z='Z�loop =e2

s2W

1

4c2w

�d4p

(2�)4(�p+ q � q)�(p� q + q)�

�ig��p2 �m2

Z

i

(p� q)2 �m2Z

(7.200)

=� e2

s2W

m2Z

4m2W

�d4p

(2�)4p2�

p2 �m2Z

��(p� q)2 �m2

Z

�=� e2

s2W

m2Z

4m2W

�d4p

(2�)41

2

�p2 �m2

Z

�+�(p� q)2 �m2

Z

�� q2 + 2pq�p2 �m2

Z

��(p� q)2 �m2

Z

�=� 1

2

e2

s2W

m2Z

4m2W

�d4p

(2�)4

�2

p2 �m2Z

+2pq � q2�

p2 �m2Z

��(p� q)2 �m2

Z

��

�� e2

s2W

m2Z

4m2W

i

16�2�2 + � � � ;

and

�i�W�='��loop =� 2e2

s2W

m2W

4m2W

�d4p

(2�)4(�p+ q � q)�(p� q + q)�

�ig��p2 �m2

W

i

(p� q)2 �m2W

(7.201)

=� 2e2

s2W

m2W

4m2W

�d4p

(2�)4p2�

p2 �m2W

��(p� q)2 �m2

W

�=� e2

s2W

m2W

4m2W

�d4p

(2�)4

�2

p2 �m2W

+2pq � q2�

p2 �m2W

��(p� q)2 �m2

W

��

�� 2e2

s2W

m2W

4m2W

i

16�2�2 + � � � :

Chiral Symmetry Breaking of QCD

By diagonalization of the mixing terms in Eq. (2.154) we �nd the masses of the lightest eight pseudoscalar

mesons in QCD. By diagonalization, we obtain

0 =det

0@ B0

2 (mu +md)�M2 B0p3(mu �md)

B0p3(mu �md)

B0

6 (mu +md + 4ms)�M2

1A (7.202)

=

�B0

2(mu +md)�M2

��B0

6(mu +md + 4ms)�M2

�� B2

0

3(mu �md)

2

=�M2�2 �M2 2B0

3(mu +md +ms) +

B20

12(mu +md)(mu +md + 4ms)� B2

0

3(mu �md)

2 )

M2 =1

2

�2B0

3(mu +md +ms)�s

4B20

9(mu +md +ms)2 � 4

�B20

12(mu +md)(mu +md + 4ms)� B2

0

3(mu �md)2

��

=B0

3(mu +md +ms)�B0

s1

36(m2

u + 2mumd � 4mums +m2d � 4mdms + 4m2

s) +1

3(mu �md)2

�

=B0

3(mu +md +ms)�B0

s1

36(2ms �mu �md)2 +

1

3(mu �md)2

�

=B0

3(mu +md +ms)� B0

6(2ms �mu �md)

�1 +

1

6

(mu �ms)2

136 (2ms �mu �md)2

�+O�(mu �ms)

3�

=2B0

3(mu +md +ms)� B0

3(2ms �mu �md)�B0

(mu �ms)2

2(2ms �mu �md)+O�(mu �ms)

3�

Page 163 of 193


After diagonalization of the mixing terms of the �elds �3 and �8, we get the masses

M2�� =(mu +md)B0; M2

K� = (mu +ms)B0; M2K0 = (md +ms)B0;

M2�0 =

�mu +md � (mu �md)

2

2(2ms �mu �md)

�B0 +O

�(mu �md)

3�; (7.203)

M2� =

�mu +md + 4ms

3+

(mu �md)2

2(2ms �mu �md)

�B0 +O

�(mu �md)

3�:

Page 164 of 193


Appendix C-3: Minimal Walking Technicolor

Rewriting of the kinetic Lagrangian in terms of Q vector

The �rst term of the kinetic Lagrangian in Eq. (3.10) with the partial derivative can be written as

i�QyAL

�_�;a

�� _��@�QAL;�;a (7.204)

=i�UyL�_�;a

�� _��@�UL;�;a + i�DyL�_�;a

�� _��@�DL;�;a + i�~UyL�_�;a

�� _��@� ~UL;�;a + i�~DyL�_�;a

�� _��@� ~DL;�;a

=i�UyL�_�;a

�� _��@�UL;�;a + i�DyL�_�;a

�� _��@�DL;�;a + i�UyR��;a

�� _�@�U_�;aR + i

�DyR��;a

�� _�@�D_�;aR ;

which is equal to the �rst part of the covariant derivative in Eq. (3.9) as expected. At the �rst equal sign

there have been summed over the index A = 1; : : : ; 4 between the two Q vectors (given in Eq. (3.7)). At

the second equal sign we have rewritten the two last terms in following way:

i�~UyL�_�;a

�� _��@� ~UL�;a =i��UyR��;b��

V yba" _� _�� _��@�

�Vac"��

�U

_�;cR

��(7.205)

=� i��UyR��;b��

�bc��_��@�

�U

_�;cR

��= i�UyR��;b�

��_��

��@�U

_�;bR

=i�UyR��;a

�� _�@�U_�;aR and

i�~DyL�_�;a

�� _��@� ~DL;�;a =i�DyR��;a

�� _�@�D_�;aR ; (7.206)


�~UyL�_�;a

=�~UL;�;a

�y=�Vab"��

�U�R��;b�y

=�Vab"��

�U

_�;bR

��y=��UyR��;b��

(Vab)y�"��y = ��UyR��;b��V yba" _� _�;

(7.207)

and that " _� _�� _��"�� = �� _�� . In the second last step in Eq. (??), we have taken the complex conjugation

of it which does not change anything because the Lagrangian is real. In the last step, we used the

hermiticity of the matrices �� = (I; ~�) which gives that

�� _�

=�� _�

�y=��_��

�� ) ��_��

��= ��

� _�: (7.208)

The second term in the covariant derivative in Eq. (3.10) can be written as

�gTCQyAL; _�;a�� _��Ai�Ti;abQAL;�;b =� gTCUyL; _�;a�� _��Ai�T

i;abUL;�;b

� gTC�Vab"��(U

�R)�;b�y

�� _��Ai�Tiad

�Vdc"��(U

�R)�;c�

+ (U $ D)

=� gTCUyL; _�;a�� _��Ai�Ti;abUL;�;b

� gTC(UyR)�;b�� _�Ai�T

bci (UR)

_�;c + (U $ D);

(7.209)


Page 165 of 193


� gTC�Vab"��(U

�R)�;b�y

�� _��Ai�Tiad

�Vdc"��(U

�R)�;c�

(7.210)

= �gTC((U�R)y) _�;b" _� _�(Vab)y�� _��Ai�T

adi Vdc"��(U

�R)�;c

= +gTC((U�R)y)

_�;b��_��Ai�(Vab)

yT adi Vdc(U�R)�;c

= gTC(UyR)�;b��

� _�Ai�

�(Vab)

yT adi Vdc

��(UR)

_�;c

= �gTC(UyR)�;b�� _�Ai�T

bci (UR)

_�;c;

where we have used in the last step, that if we have a unitary transformation,

V �1TiV = �(Ti)�; (7.211)

then for V = I such that Ti = �(Ti)� for every i we have that the representation R is real. If V 6= I, we


is complex. If Ti is in the complex representation in Eq. (7.210), then we have not the SU(4) symmetry,

because we can not perform the last step in Eq. (7.210). We have instead only the SU(2)L SU(2)R

symmetry as in the kinetic Lagrangian in Eq. (3.9). In our case, the gauge group is a SU(2) gauge group,

which is in the pseudoreal representation, therefore there is a SU(4) symmetry.

By inserting Eq. (7.204) and Eq. (7.209) into Eq. (3.10), we obtain that

LK =iQyAL; _�;a�� _��Dab

� QAL;�;b = iUyL; _�;a��

� _��Dab� UL;�;b + iDyL; _�;a��

� _��Dab� DL;�;b+ (7.212)

i�Vab"��(U

�R)�;b�y�� _��Dad

�

�Vdc"��(U

�R)�;c�+ i�Vab"��(D

�R)�;b�y�� _��Dad

�

�Vdc"��(D

�R)�;c�

=iUyL; _�;a�� _��Dab

� UL;�;b + iDyL; _�;a�� _��Dab

� DL;�;b + iUy;�R;a�� _�D

ab� U

_�R;b + iDy;�R;a�

�� _�D

ab� D

_�R;b;

where Dab� = @��

ab + igTCAi�T

iab is the covariant derivative. Then we have shown that the kinetic

Lagrangian in Eq. (3.9) can be written as

LK = iQyAL; _�;a�� _��Dab

� QAL;�;b: (7.213)

The condensate in SO(4) and Sp(4)

In the following we will derive an expression of the mass term in term of the SU(4) vector Q. We have

that

�2[ �URUL + �DRDL] =� 2�Uy�aR UL�;a +Dy�;aR DL;�;a

�= �Uy�;aR UL;�;a � Uy�;aR UL;�;a + (U $ D)

=� (U�R)�;bUL;�;a"��V ab � (U�R)�;aU

�;aL + (U $ D)

=Q1L;�;aQ

3L;�;b"

��V ab �Q3L;�;aQ

1L;�;b"

��V ab +Q2L;�;aQ

4L;�;b"

��V ab� (7.214)

Q4L;�;aQ

2L;�;b"

��V ab = QAL;�;aQBL;�;b"

��V abE�AB = QAL;�;aQB;�;aL E�AB

=QTE�Q;

Page 166 of 193


where

E� =

0@ 0 1

�1 0

1A =

0BBBBBB@

0 0 1 0

0 0 0 1

�1 0 0 0

0 �1 0 0

1CCCCCCA: (7.215)

We have E+ in the mass term, if the matrix V ab is symmetric (V ab = V ba), and E� if it is antisymmetric

(V ab = �V ba). From Eq. (36.19) in Ref. [10], we have used that the left- and right-handed Dirac spinors

can be written as

UL =

0@ UL;�;a

0

1A and UR =

0@ 0

U _�;aR

1A (7.216)

and from Eq. (36.21) in Ref. [10] that the adjoint Dirac spinors is de�ned as �QL � QyL�, where

� =

0@ 0 � _�_�

�� 0

1A : (7.217)

Thus, the left- and right-handed adjoint Dirac spinor is

�UL =UyL� =�

(UyL) _�;a 0�0@ 0 � _�_�

�� 0

1A =

�0 (UyL) _�;a

�; (7.218)

and

�UR =UyR� =�

0 (UyR)�;a

�0@ 0 � _�_�

�� 0

1A =

�(UyR)

�;a 0�: (7.219)

From this we obtain that

�URUL =�Uy�;aR 0

�0@ UL;�;a

0

1A = (UyR)

�;aUL;�;a: (7.220)

The �rst equality in Eq. (3.12) follows from that we have used Eq. (7.220). The third equality gives

either plus or minus in the second term because by switching �� to �

� introduces an extra minus sign

and together with switching aa to a

a introduces either +1 or �1 (+1 if V ab is symmetric and �1 if V ab

is antisymmetric). I.e. that

Uy�;aR UL;�;a =U��;aR UL;�;a = "��V abU�R;�;bUL;�;a = �"��V abU�R;�;bUL;�;a

=� "��V baU�R;�;bUL;�;a = �U�R;�;bU�;bL = �U�R;�;aU�;aL :(7.221)

In the following, we will show that the condensate hQTE�Qi = �2h �URUL+ �DRDLi is invariant underSO(4). The representation of SU(4) in Eq. (7.1) in Appendix A can be inserted into the criterion in Eq.

(3.15). By inserting the Sa generators, we obtain

Page 167 of 193


(Sa)TE� + E�Sa =

0@ AT B�

BT �A

1A0@ 0 I

�I 0

1A+

0@ 0 I

�I 0

1A0@ A B

By �AT

1A

=

0@ �B� AT

�A BT

1A+

0@ By �AT

�A �B

1A

=

0@ �B� AT

�A BT

1A+

0@ �B� �AT

�A �B

1A

=

0@ �B� �B� 0

0 �B �B

1A ; a = 1; : : : ; 6;

(7.222)

and by inserting the Xi generators, we get

(Xi)TE� + E�Xi =

0@ CT D�

DT C

1A0@ 0 I

�I 0

1A+

0@ 0 I

�I 0

1A0@ C D

Dy CT

1A

=

0@ �D� CT

�C DT

1A+

0@ Dy CT

�C �D

1A =

0@ �D� CT

�C D

1A+

0@ D� CT

�C �D

1A

=

0@ �D� +D� 2CT

�2C D �D

1A ; i = 1; : : : ; 9;

(7.223)

where we have used that BT = �B and DT = D. We have that B = 0 for Sa when a = 1; : : : ; 4, A = 0

for Sa when a = 5; 6, D = 0 for Xi when i = 1; 2; 3, and C = 0 for Xi when i = 4; : : : ; 9. Therefore, for

E+ the relation in Eq. (3.15) is maintained for the generators Sa but not for Xi. For E� the relation

is maintained for the generators Sa where a = 1; : : : ; 4 and for Xi where i = 4; : : : ; 9. I.e. that the

condensate in Eq. (3.12) is invariant under SO(4) transformations for E+ and invariant under Sp(4)

transformations for E�.

Low Energy Theory for MWT

In the following we show that the second term of the M matrix in Eq. (3.24) is invariant under SO(4)

transformations. This term can be shown to be invariant if the condition in Eq. (3.26) is maintained and

by using the generators of SU(4) in Appendix A. We obtain that

SbXaE +XaESbT =

0@ Ab Bb

Bby �AbT

1A0@ Ca Da

Day CaT

1A0@ 0 1

1 0

1A+

0@ Ca Da

Day CaT

1A0@ 0 1

1 0

1A0@ AbT Bb�

BbT �Ab

1A =

0@ AbDa +DaAb� +BbCa� �CaBb AbCa �CaAb +BbDa� +DaBa�

�Ab�Ca� +Ca�Ab� �Bb�Da �Da�Bb �Ab�Da� �Da�Ab �Bb�Ca +Ca�Bb�

1A = 0;

(7.224)

and therefore the second term of the M is invariant under SO(4).

Page 168 of 193


Yukawa Interactions

The U(1)V charge of the Mo� in Eq. (3.85) is zero, because

S4Mo� +Mo�S4T =S4

�� + i�

2+p2(i�i + ~�i)Xi

�E+ +

�� + i�

2+p2(i�i + ~�i)Xi

�E+S4T

=� + i�

2

0@ 0 1

1 0

1A+

p2(i�i + ~�i)

0@ 0 � i

��aT 0

1A+

� + i�

2

0@ 0 �1�1 0

1A+

p2(i�i + ~�i)

0@ 0 �� i

�aT 0

1A = 0;

(7.225)

where i = 1; 2; 3, and we have used the vacuum E+ in Eq. (3.13) and the Sa matrices with a = 1; : : : ; 4

in Eq. (7.2) in Appendix A.

Weinberg Sum Rules and Electroweak Parameters

The determination of the Weinberg sum rules (WSRs) is done as follows

1

�

� 1

0

dsIm�(s)

s+Q2� Q�6 ) 1

�

� 1

0

dsIm�(s)

1 + s=Q2� Q�4 ) 1

�

� 1

0

dsIm�(s)�1�Q�2s� � Q�4: (7.226)

Therefore, the �rst and the second WSR are

1

�

� 1

0

dsIm�(s) = 0;1

�

� 1

0

dssIm�(s) = 0: (7.227)

We combine the two WSRs, which are

f2V � f2A = f2� ; (7.228)

and

f2Vm2V � f2Am2

A ' a8�2

d(R)f4� : (7.229)

We can rewrite these WSRs as follows

f2V = f2� + f2A;

f2�m2V + f2Am

2V � f2Am2

A = a8�2

d(R)f2�f2� ;

(7.230)

which give by combining them

m2V �m2

A 'f2�f2A

�a

8�2

d(R)f2��m2

V

�: (7.231)

Page 169 of 193


Appendix C-4: Composite Higgs Dynamics

The Space of Vacua

In this appendix, we will derive the most general form of the vacua. We can write the fermion condensate

as

hABi / �AB ; (7.232)

where we must have that

�T = ��; (7.233)

because hBAi = �hABi ) �BA = ��AB . We assume that the vacuum is preserved under

Sp(4) 2 SU(4) transformations, i.e. that V �0VT = �0 (V 2 Sp(4)), where

�0 =

0@ 0 12�2

12�2 0

1A) �0�

y0 = 1: (7.234)

This is satis�ed if we have that

�y� = 1; (7.235)

because we have ��y = U�0UTU��y0U

y = U�0�y0U

y = 1, when we can construct all vacua by rotating

the vacuum �0 with an SU(4) transformation as follows � = U�0UT (U 2 SU(4)) and �0�

y0 = 1.

The Pfa�an is given as

Pf(�) = 14�abcd�

ab�cd; (7.236)

which is invariant under SU(4) transformations and transforms as Pf(�) ! Pf(�)� under CP trans-

formations, because the condensate transforms as � ! �y under CP . We assume that the vacua is

preserved under CP , and therefore the Pfa�an is real. We choose the normalization of the condensate

such that

Pf(�) = �1: (7.237)

We have that the Eq. (7.233), (7.235) and (7.237) are invariant under SU(4) and CP transformations.

According to these constraints, we can construct the most general condensate, which can be written

� =

0@ a� c

�cT b�

1A ; (7.238)

because it satis�es the antisymmetric condition in Eq. (7.233)

�T =

0@ �a� �c

cT �b�

1A = �

0@ a� c

�cT b�

1A = ��; (7.239)

and the condition in Eq. (7.235)

�y� =

0@ �a�� c�

cy �b��

1A0@ a� c

�cT b�

1A =

0@ a�a+ c�cT �a��c� c�b�

cya�+ b��cT cyc+ b�b

1A = 1: (7.240)

Page 170 of 193


The expression above gives the equations

ccy + jaj2 12�2 = ccy + jbj2 12�2 = 12�2 (7.241)

acy�+ b��cT = 0: (7.242)

The Eq. (7.241) implies c = ru, where r is a real number, u = ei is unitary and is a real number given

by

ccy + jaj2 12�2 = ccy + jbj2 12�2 = 12�2 )r212�2 + jaj2 12�2 = r212�2 + jbj2 12�2 = 12�2 ,r2 = 1� jaj2 = 1� jbj2 :

(7.243)

Therefore, the Eq. (7.242) implies

acy�+ b��cT = 0) aruy�+ b��ruT = 0) ae�i + b�ei ,ae�i = �(be�i )�:

(7.244)

We have that r2 > 0, which leads to that r = sin �, when jaj = cos � for 0 � � � �. Therefore, we have

that

a =cos �ei� and

c =ru = rei 12�2 = sin �ei 12�2:(7.245)

By using Eq. (7.242) we can derive the expression of b as follows

ae�i = �b�ei ) cos �ei�e�i = �b�ei ) cos �ei� = �b�ei )b = � cos �e�i�ei ;

(7.246)

where we have de�ned � � � � . Thus, the elements of the matrix of the general condensate in Eq.

(7.238) can be written as

a� =cos �ei�� = ei ei� cos ��

c =ei sin �12�2

�cT =� ei sin �12�2b� =� ei e�i� cos ��

(7.247)

We can conclude that the most general condensate is

� = ei

0@ ei� cos �� sin �12�2

� sin �12�2 �e�i� cos ��

1A : (7.248)

The most general condensate satis�es Eq. (7.233), (7.235), and the P��an in Eq. (7.237) which gives

Pf(�) = 14�abcd�

ab�cd = e2i 14 [� cos2 � � cos2 � � sin2 � � sin2 � � cos2 � � cos2 � � sin2 ��sin2 � � cos2 � � sin2 � � cos2 � � sin2 �] = �e2i 144

�cos2 � + sin2 �

�= �e2i ;

(7.249)

and therefore we have that

Pf(�) = �e2i = �1) = 0: (7.250)

Page 171 of 193


So the most general vacuum is (as Eq. (A.17) in Appendix A in Ref. [26])

� =

0@ ei� cos �� sin �12�2

� sin �12�2 �e�i� cos ��

1A : (7.251)

An extra information is that the sign of the block o�-diagonal entry can be changed with a SU(2)W (or

U(1)Y) transformation, and therefore the angle � is in the range 0 � � � �.

Expansion of the Kinetic Term

In this subsection we will expand the kinetic-gauge term of �:

f2Tr[(D��)yD��]; (7.252)

where the �elds h and � are parameritized as

� = eif(hY 4+�Y 5); (7.253)

and the covariant derivative is

D�� = @� � ig[G�(y)� + �GT� (y)]; (7.254)

and

gG�(YV ) = gW a�L

a + g0B�Y = gW a�S

a + g0B�S6 (7.255)

We can expand � for small values of the �elds as follows

� =eif(hY 4+�Y 5) � �0

=

�1 +

i

f(hY 4 + �Y 5)� 1

16f2(h2 + �2)� i

48f3(h3Y 4 + �3Y 5 + h�2Y 4 + �h2Y 5)+

1

24 � 64f4 (h4 + �4 + 2h2�2) +

1

64 � 120f5 (h5Y 4 + h�4Y 4 + 2h3�2Y 4 + �h4Y 5 + �5Y 5+

2�3h2Y 5) +O(f�6)�� 0;

(7.256)


(Y 4)2 = (Y 5)2 =1

8

0@ 12 0

0 12

1A ; Y 4Y 5 =

1

8

0@ �is�12 c��

2

�c��2 is�12

1A = �Y 5Y 4: (7.257)

where the broken generators Y 4 and Y 5 are from Eq. (4.15).

Firstly, we can expand the pure kinetic term of � as follows

f2Tr[@��y@��] =Tr

�@�

�� i(hY 4 + �Y 5)� 1

16f(h2 + �2) +

i

48f2(h3Y 4 + �3Y 5 + h�2Y 4+

�h2Y 5)

�@��i(hY 4 + �Y 5)� 1

16f(h2 + �2)� i

48f2(h3Y 4 + �3Y 5+

h�2Y 4 + �h2Y 5)

��0�

y0

�+O(f�3)

(7.258)

Page 172 of 193


=Tr

�1

8(@�h@

�h+ @��@��) +

1

64f2(h@�h+ �@��)

2 � 1

24 � 8f2 (@�h@�(h3)+

@�h@�(h�2) + @��@

�(�3) + @��@�(�h2)

�+O(f�3)

=1

2(@�h)

2 +1

2(@��)

2 +1

48f2[�(h@�� @�h)2] +O(f�3):

Thereafter, we can expand the mix terms, which is

f2Tr

�@��

y(�ig[G�(y)� + �GT� (y)]) + ig[G�(y)� + �GT� (y)]@��

�= 0: (7.259)

Finally, we can expand the gauge interaction terms as follows

f2Tr

�� ig[G�(y)� + �GT� (y)ig[G�(y)� + �GT� (y)

�=

�2g2W+

� W�� + (g2 + g02)Z�Z�

��f2s2� +

s2�f

2p2h

�1� 1

12f2(h2 + �2)

�+

1

8

�c2�h

2 � s2��2��

1� 1

24f2(h2 + �2)

��+O(f�3):

(7.260)

We can collect these three terms such that we get the kinetic term of � which includes the interactions

with the gauge bosons via minimal coupling, which yields

f2Tr[(D��)yD��] =

1

2(@�h)

2 +1

2(@��)

2 +1

48f2[�(h@�� @�)2]+

�2g2W+

� W�� + (g2 + g02)Z�Z�

��f2s2� +

s2�f

2p2h

�1� 1

12f2(h2 + �2)

�+

1

8

�c2�h

2 � s2��2��

1� 1

24f2(h2 + �2)

��+O(f�3);

(7.261)

where the covariant derivative of � is

D�� = @�� igW a� (S

a�+ �SaT )� ig0B�(S6�+ �S6T ): (7.262)

Covariant Derivative

In this section, the covariant derivative of the techniquark bilinears � � QQT will be derived from that

the kinetic term Tr((D��)yD��) should be invariant under the gauge transformations. This is the case

if the covariant derivative transforms as

D��! u(D��)uT ; (7.263)

where u = exp(i�a(x)T a) is a gauge transformation. We have that the � and the gauge �elds A� � Aa�Ta

transform as follows

�! u�uT and

A� = Aa�Ta ! uAa�T

auy +i

gu(@�u

y);(7.264)

where g is the coupling constant for the gauge �eld A�. Thus, we have that the covariant is and transforms

Page 173 of 193


as

D�� =@�� ig�A��+ �AT�

�!(D��)

0 =@�(u�uT )� ig��uA�u

y +i

gu(@�u

y)�u�uT + u�uT

�u�AT�u

T +i

g(@�u

�)uT��

=u(@��)uT + i(@��

a)T au�uT + iu�(@��a)T aTuT � ig

�uA��u

T + u�AT�uT+

1

gu(@��

a)T auyu�uT +1

gu�uT (@��

a)T aTu�uT�

=u(@��)uT + i(@��

a)T au�uT + iu�(@��a)T aTuT � iguA��uT � igu�AT�uT�

iu(@��a)T a�uT � iu�uT (@��a)T aT

=u�@�� ig

�A��+ �AT�

��uT = u(D��)u

T :

(7.265)

Therefore, we have that the covariant derivative in Eq. (4.19) has the form

D�� = @�� igW a�

�Sa�+ �SaT

�� ig0B��S6�+ �S6T�; (7.266)

where the generators S1;2;3 are identi�ed with the electroweak generators for SU(2)W and S6 for U(1)Y.

Expansion of One-Loop Potentials

In this subsection we expand the one-loop potentials of the SU(2) and U(1) gauge bosons, top quark

and the explicit breaking term of SU(4). We start to expand the one-loop potential of the SU(2) gauge

bosons, which according to Eq. (4.32) has the form


Tr�Si � � � (Si � �)��: (7.267)

We need to expand the exponential parameterization of the not absorbed pNGBs as follows

� =eif(hY 4+�Y 5) =

�1 +

i

f(hY 4 + �Y 5)� 1

16f2(h2 + �2) +O(f�3)

�� 0; (7.268)

where we use the matrices

�0 =

0@ ic��

2 s�

�s� �ic��2

1A ; Y 4 =

1

2p2

0@ 0 �2

�2 0

1A ; Y 5 =

1

2p2

0@ c� �is��2

is��2 �c�

1A ;

Sa =1

2

0@ �a 0

0 0

1A ; S6 =

1

2

0@ 0 0

0 ��3

1A :

(7.269)

Therefore, we have that

Tr�Si � �(Si � �)�� = Tr

�Si�1 +

i

f

�hY 4 + �Y 5

�� 1

16f2

�h2 + �2

��0S

i�

�1� i

f

�hY 4� + �Y 5�

�� 1

16f2

�h2 + �2

��0

�+ � � � =

Tr�Si�0S

i��0�+i

fhTr

�SiY 4�0S

i��0�� i

fhTr

�Si�0S

i�Y 4��0�+

(7.270)

Page 174 of 193


i

f�Tr�SiY 5�0S

i��0�� i

f�Tr�Si�0S

i�Y 5��0�+

1

f2h2Tr

�SiY 4�0S

i�Y 4��0��

1

8f2h2Tr

�Si�0S

i��0�� 1

8f2�2Tr

�Si�0S

i��0�+

1

f2�2Tr

�SiY 5�0S

i�Y 5��0�+

i

f�Tr�SiY 5�0S

i��0�� i

f�Tr�Si�0S

i�Y 5��0�+ : : : :

In the following it is needed that

Tr[��1�2�1�2] = Tr[��2�2�2�2] = Tr[��3�2�3�2] = �2)Tr[��i �2�i�2] = �6;

(7.271)

where there is a sum over i = 1; 2; 3. The various traces above are

Tr�Si�0S

i��0�= �c

2�

4Tr[��i �2�i�2] =

3c2�2;

Tr�SiY 4�0S

i��0�= � i

8p2c�s�Tr[�

�i �2�i�2] =

3i

4p2c�s�;

Tr�Si�0S

i�Y 4��0�=

i

8p2c�s�Tr[�

�i �2�i�2] = �

3i

4p2c�s�;

Tr�SiY 5�0S

i��0�= � 1

8p2c�Tr[�

�i �2�i�2] =

3

4p2c�;

Tr�Si�0S

i�Y 5��0�= � i

8p2c�s

2�Tr[�

�i �2�i�2] =

3i

4p2c�s

2�

Tr�SiY 4�0S

i�Y 4��0�= � 1

32s2�Tr[�

�i �2�i�2] =

3

16s2�;

Tr�SiY 5�0S

i�Y 5��0�=

1

32Tr[��i �2�i�2] = �

3

16(s2� + c2�):

(7.272)

Thus, the one-loop potential of the SU(2) gauge bosons can be expanded as follows


Tr�Si � � � (Si � �)��

=� Cgg2f4�3

2c2� �

3

2p2fc�s�h� 3

16f2(c2�h

2 � s2��2) + : : :

�:

(7.273)

Now, we will expand the potential of the U(1) gauge boson, which is

VU(1) =� Cgg02f4Tr�S6 � � � (S6 � �)�� : (7.274)

Thus, the trace can be expanded as follows

Tr�S6 � � � (S6 � �)�� = Tr

�S6�1 +

i

f

�hY 4 + �Y 5

�� 1

16f2

�h2 + �2

��0S

6�

�1� i

f

�hY 4� + �Y 5�

�� 1

16f2

�h2 + �2

��0

�+ � � � =

Tr�S6�0S

6��0�+i

fhTr

�S6Y 4�0S

6��0�� i

fhTr

�S6�0S

6�Y 4��0�+

i

f�Tr�S6Y 5�0S

6��0�� i

f�Tr�S6�0S

6�Y 5��0�+

1

f2h2Tr

�S6Y 4�0S

6�Y 4��0��

1

8f2h2Tr

�S6�0S

6��0�� 1

8f2�2Tr

�S6�0S

6��0�+

1

f2�2Tr

�S6Y 5�0S

6�Y 5��0�+

(7.275)

Page 175 of 193


i

f�Tr�S6Y 5�0S

6��0�� i

f�Tr�S6�0S

6�Y 5��0�+ : : : ;

where the traces in the expression above are

Tr�S6�0S

6��0�= �c

2�

4Tr[��3�2�3�2] =

c2�2;

Tr�S6Y 4�0S

6��0�= � i

8p2c�s�Tr[�

�3�2�3�2] =

i

4p2c�s�;

Tr�S6�0S

6�Y 4��0�=

i

8p2c�s�Tr[�

�3�2�3�2] = �

i

4p2c�s�;

Tr�S6Y 5�0S

6��0�= � 1

8p2c�Tr[�

�3�2�3�2] =

1

4p2c�;

Tr�S6�0S

6�Y 5��0�= � i

8p2c�s

2�Tr[�

�3�2�3�2] =

i

4p2c�s

2�

Tr�S6Y 4�0S

6�Y 4��0�= � 1

32s2�Tr[�

�3�2�3�2] =

1

16s2�;

Tr�S6Y 5�0S

6�Y 5��0�=

1

32Tr[��3�2�3�2] = �

1

16(s2� + c2�):

(7.276)

Thus, the potential of the U(1) gauge boson can be expanded as follows

VU(1) =� Cgg02f4Tr�S6 � � � (S6 � �)��

=� Cgg02f4�1

2c2� �

1

2p2fc�s�h� 1

16f2(c2�h

2 � s2��2) + : : :

�:

(7.277)

With same procedure the potentials of the top quark and the explicit breaking term of SU(4) can be

expanded as follows

Vtop =� Cty02t f42X

�=1

[Tr(P��)]2

=� Cty02t f4�s2� +

1p2fc�s�h+

1

8f2(c2�h

2 � s2��2) + : : :

�;

(7.278)

and

Vm =Cmf4Tr(�B � �)

=Cmf4

��4c� + 1p

2fs�h+

1

4f2c�(h

2 + �2) + : : :

�:

(7.279)

Page 176 of 193


Appendix C-5: Partially Composite Higgs

In this appendix, we will derive the various expressions and equations in Ref. [3], which describes a

partially composite Higgs model with one fundamental Higgs doublet.

E�ective Lagrangian building block Sp(4) transformation

We have that the �rst term in the fundamental Higgs potential with the technifermons in Eq. (5.7)

transforms under U 2 SU(4) as

�T �C�1(M + �)! �TUT �C�1(M 0 + �0)U�; (7.280)

which is invariant under SU(4) if the matrix transforms as M + �! U�(M + �)Uy. Therefore, we have

that the building block in Eq. (5.23) transforms under Sp(4) as

�� = �T (M + �)�� h.c.!V ��TV TV �(M + �)V yV �V yV �V T � h.c. =

V ��T (M + �)��V T � h.c. = V ��V T ;(7.281)

where the element V 2 Sp(4).

The EW scale

In this subsection we derive the relation for the EW scale

v2EW = f2 sin2 � + v2: (7.282)

We look only at the mass of the W� bosons by setting the rest of the �elds equal zero, and therefore we

have � = 1 according to Eq. (5.18), i.e. that C� = A� from Eq. (5.19) and Eq. (5.20). In this case from

Eq. (5.6) we get

A� =

0@ g2W

a�12�

a 0

0 0

1A ; (7.283)

and therefore we have

C?� = Tr [A�Xa]Xa =1

2p2

"Tr

240@ g2W

i��

i 0

0 0

1A0@ �s��1 c��3

c��3 �s��1

1A35X1+

Tr

240@ g2W

i��

i 0

0 0

1A0@ s��2 ic�

�ic� �s��2

1A35X2 + Tr

240@ g2W

i��

i 0

0 0

1A0@ s��3 c��1

c��1 s��3

1A35X3+

Tr

240@ g2W

i��

i 0

0 0

1A0@ 0 �2

�2 0

1A35X4 + Tr

240@ g2W

i��

i 0

0 0

1A0@ c� �is��2

is��2 �c�

1A35X5

#

=g2

2p2

�� s�W i

�Tr[�i�1]X1 � s�W i

�Tr[�i�2]X2 + s�W

i�Tr[�

i�3]X3 + 0 + c�Wa�Tr[�

a]

�

=1p2g2s�

��W 1�X

1 �W 2�X

2 +W 3�X

3�:

(7.284)

Page 177 of 193


The �rst term in the TC e�ective Lagrangian in Eq. (5.24) gives the following mass term for the W�

bosons

L(2) = f2

2Tr[C?� C

?�] + � � � = f2

4g22s

2�Trh(�W 1

�X1 �W 2

�X2)(�W 1�X1 �W 2�X2)

i+ : : :

=f2

4g22s

2�

hW 1�W

1�Tr[X1X1] +W 2�W

2�Tr[X2X2]i+ � � � = f2

4g22s

2�W

+� W

�� + : : :

= m2WW

+� W

�� + : : :

(7.285)

Thus, we have that m2W = g2v

2=4 resulting in the contribution f sin � to the electroweak VEV. The Higgs

gauge-kinetic term in Eq. (5.7)1

2Tr�(D�H)yD�H

�(7.286)

yield also with a mass term to the W� bosons, where

H =1p2

0@ v 0

0 v

1A and D� = �i

0@ g2W

a�12�

a 0

0 0

1A : (7.287)

Thus, we get that

1

2Tr�(D�H)yD�H

�=1

4

240@ g2W

a�12�

a 0

0 0

1A0@ v 0

0 v

1A0@ g2W

b�12�

b 0

0 0

1A0@ v 0

0 v

1A35+ : : :

=v2

16Tr[g22W

a�W

b��a�b] =v2

16g22W

a�W

b�2�ab =v2g224

W+� W

�� + : : : :

(7.288)

This mass term contributes with v to the electroweak VEV. When we combine these two mass terms,

then we get that the EW scale is

v2EW = f2 sin2 � + v2: (7.289)

The Fundamental Higgs Potential

In this subsection, the O(p2) potential is derived from second TC Lagrangian term in Eq. (5.24) and

the fundamental Higgs potential in Eq. (5.7). Thereafter, the potential is minimized. We de�ne that

m12 = m1 +m2, �UD = �U + �D and mUD = mU +mD = v(�U + �D)=p2 = v�UD=

p2. The second

TC Lagrangian term in Eq. (5.24)

4�f3Z2Tr(�+) (7.290)

contributes to the O(p2) potential. When we set all �elds to zero, then according to Eq. (5.3) and Eq.

(5.10) we obtain the matrices

� =1

2

0BBBBBB@

0 ei� cos � sin � 0

�ei� cos � 0 0 sin �

� sin � 0 0 �e�i� cos �0 � sin � e�i� cos � 0

1CCCCCCA; (7.291)

Page 178 of 193


M + � =1

2

0BBBBBB@

0 m1 ��Uv�=p2 0

�m1 0 0 ��Dv=p2

�Uv�=p2 0 0 �m2

0 �Dv=p2 m2 0

1CCCCCCA

+ : : : : (7.292)

We have � = 1, because we have set all �elds to zero. Thus, we get

�+ =�T (M + �)��+ h.c.

=

0BBBBBB@

�m1ei� cos �

�m1ei� cos �

�m2e�i� cos �

�m2e�i� cos �

1CCCCCCA

+

0BBBBBB@

�Uv�p2sin �

�Dvp2sin �

�Uv�p2sin �

�Dvp2sin �

1CCCCCCA

+ : : : :

(7.293)

Thus, the constant part of the TC Lagrangian term in Eq. (7.290) is

4�f3Z2Tr(�+) =4�f3Z2

��Uv

�p2

+�Dvp

2

�sin � � �m1e

i� +m2e�i�� cos ��

=� 8�f3Z2


p2i;

(7.294)

where we have set v� = v and � = 0.

We obtain also a contribution to the O(p2) potential from the fundamental Higgs potential in Eq.

(5.7), which is

m2H jHj2 + �hjHj4 = 1

2m2Hv

2 +1

4�hv

4 + : : : ; (7.295)

where

H =1p2

0@ 0

v

1A+ : : : : (7.296)

Thus, the total O(p2) potential is

V(2)e� = 8�f3Z2


p2i+m2H

2v2 +

�h4v4: (7.297)

We minimize this O(p2) potential

@V(2)e�

@�=8�f3Z2

h�m12 sin � � �UDv cos �=

p2i= 0 (7.298)

@V(2)e�

@v=� 8�f3Z2�UD sin �=

p2 +m2

Hv + �hv3 = 0 (7.299)

From the �rst equation we obtain

tan � = �mUD

m12; (7.300)

Page 179 of 193


and from the second equation we can obtain an expression of the Higgs self-coupling

�h =4p2�Z2f

3 sin � �m2Hv

v3: (7.301)

Mass Matrices

Page 180 of 193


The Mathematica script above can �nd the mass terms that contribute to the masses of the composite

�elds from the TC e�ective Lagrangian term 4�f3Z2Tr(�+).

In this subsection, the mass matrices of the neutral Higgs in the basis (�h; �4), the charged scalar in

the basis (�+h ; �+) (��h = (�1h� i�2h)=

p2 and �� = (�1� i�2)=p2), the neutral scalar in the basis (�3; �

3)

and the mass of �5 are derived from the terms of the fundamental Higgs potential �m2H jHj2��hjHj4 in

Page 181 of 193


Eq. (5.7) and from the TC e�ective Lagrangian term 4�f3Z2Tr(�+) in Eq. (5.24).

The Mathematica script has found that the mass term of �4 is

� 12m

2�4(�

4)2 = �4f�Z2(s�mUD � c�m12)(�4)2; (7.302)

where mUD � v(�U + �D)=p2 � v�UD=

p2 and m12 � m1 +m2. Thus, the mass of �4 is

m2�4 =16�fZ2

1

2(�m12c� +mUDs�) = 16�fZ2

1

2s�mUD =

8�fZ2�UDp2s�

v =1

s2�f2(m2

Hv2 + �hv

4)

=m2Ht

2� + �ht

2�v

2;

(7.303)


1

2(�m12c� +mUDs�) =

1

2

�mUD

t�c� +mUDs�

�=mUD

2s�; (7.304)

where we have used tan � = �mUD=m12 in Eq. (7.300). Furthermore, we use that mUD = �UDv=p2,

tan� � v=(f sin �) and Eq. (5.28), which gives

8�f3Z2�UDs�p2

= m2Hv + �hv

3 ) 8�fZ2�UDp2s�

=1

f2s2�(m2

Hv + �hv3): (7.305)

Now, we can used the Mathematica script to �nd the rest of the masses. The mass of the cross term

�4�h is

m2�4�h

= �8�f3Z2c��UD=p2 = �c� 1

fs�(m2

Hv + �hv3) = �c�t�(m2

H + �hv2); (7.306)

where the Eq. (7.305) is used again. We have two contributions to the �2h mass term, which come from

the following terms of the fundamental Higgs potential

m2H jHj2 =

1

2m2H(v + �h)

2 + � � � = 1

2m2H�

2h + : : : ;

�hjHj4 = 1

4�h(v

2 + �2h + 2�hv)2 + � � � = 3

2�hv

2�2h + : : : :

(7.307)

Thus, the mass of �h �eld is

m2�h = m2

H + 3�hv2: (7.308)

By combining these masses, the neutral Higgs mass matrix in the basis (�h; �4) can be written as

M2h =

0@ m2

H + 3�hv2 �m2

Hc�t� � �hv2c�t��m2

Hc�t� � �hv2c�t� m2Ht

2� + �hv

2t2�

1A

=m2H

0@ 1 �c�t��c�t� t2�

1A+ �hv

2

0@ 3 �c�t��c�t� t2�

1A :

(7.309)

Next, we can �nd the charged scalar mass matrix in the basis (�+h ; �+) in the same way. Thus, the

mass of �+�� is

m2�+ =8�fZ2(�c�m12) + s�mUD) = 8�fZ2

mUD

s�= 8�fZ2

�UDvp2s�

=v

f2s2�(m2

Hv + �hv3) = t2�(m

2H + �hv

2);

Page 182 of 193


and the mass of �+�� is

m2�+h��

= �8p2�f2�UDZ2 = � 1

fs�(m2

Hv + �hv3) = �t�(m2

H + �hv2): (7.310)

We have two contributions to the �+h ��h mass, which come from the following terms of the fundamental

Higgs potential

m2H jHj2 =

1

2m2H((�

1h)

2 + (�2h)2) + � � � = m2

H�+h �

�h + : : : ;

�hjHj4 = 1

4�h((�

1h)

2 + (�2h)2 + v2)2 + � � � = 1

2�h((�

1h)

2 + (�2h)2)v2 + � � � = �hv

2�+h ��h + : : : :

(7.311)

Therefore, the mass of �+h ��h is

m2�+h

= m2H + �hv

2: (7.312)

By combining the masses, the charged scalar mass matrix in the basis (�+h ; �+) can be written as

M2�+ =

0@ m2

H + �hv2 �m2



2� + �hv

2t2�

1A = (m2

H + �hv2)

0@ 1 �t��t� t2�

1A : (7.313)

The last mass matrix is the neutral scalar mass matrix in the basis (�3h; �3) that we can �nd. The

mass of �3 is

m2�3 = 8�fZ2(s�mUD � c�m12) = 8�fZ2

mUD

s�=

v

f2s2�(m2

Hv + �hv3) = t2�(m

2H + �hv

2); (7.314)

and the mass of �3h�3 is

m2�3h�3 = �8

p2�f2Z2�UD = �t�(m2

H + �hv2): (7.315)

We have two contributions to the (�3h)2 mass, which come from the following of the fundamental Higgs

potential

m2H jHj2 =

1

2m2H(v � i�3h)(v + i�3h) + � � � =

1

2m2H(�

3h)

2 + : : : ;

�hjHj4 = 1

4�h(v

2 + (�3h)2)2 =

1

2�hv

2(�3h)2 + : : : :

(7.316)

Thus, the mass of (�3h)2 is

m2�3h= m2

H + �hv2: (7.317)

Combining these masses, we can write the neutral scalar mass matrix in the basis (�3h; �3) as

M2�3 =

0@ m2

H + �hv2 �m2



2� + �hv

2t2�

1A = (m2

H + �hv2)

0@ 1 �t��t� t2�

1A : (7.318)

Finally, we can derive the mass of �5 in the same way by using the Mathematica script above. The

mass of �5 is

m2�5 = 8�fZ2(s�mUD � c�m12) = 8�fZ2

mUD

s�=

v

f2s2�(m2

Hv + �hv3) = t2�(m

2H + �hv

2): (7.319)

Page 183 of 193


The Mass Eigenstates

We will derive in this subsection the mass eigenstates and their masses from the mass matrices in the

previous subsection. We de�ne " � 3�hv2=m2

H for simplicity.

Firstly, we derive the mass eigenvalues of the neutral Higgs mass matrix

M2h =

0@ m2

H + 3�hv2 �m2

Hc�t� � �hv2c�t��m2

Hc�t� � �hv2c�t� m2Ht

2� + �hv

2t2�

1A

=m2H

0@ 1 + " �c�t�(1 + "=3)

�c�t�(1 + "=3) t2�(1 + �=3)

1A :

(7.320)

The eigenvalue equation of this mass matrix is

0 =�m2H(1 + ")�M2

� �m2Ht

2�(1 + "=3)�M2

�� c2�m4Ht

2�(1 + "=3)2

=(M2)2 �M2�m2H(1 + ") +m2

Ht2�(1 + "=3)

�+m4

Ht2�(1 + ")(1 + "=3)� c2�m4

Ht2�(1 + "=3)2;

(7.321)

which has the mass solutions

m2h1;h2 =

m2H

2

�1=c2� + "(1 + t2�=3)�r�

1=c2� + "(1 + t2�=3)�2� 4

��c2�t2�(1 + "=3)2 + t2�(1 + ")(1 + "=3)

��:

(7.322)

We assume that the mass eigenstates of the mass matrix are

h1 = c��h � s��4 and h2 = s��h + c��4: (7.323)

If this is the case, then we obtain that

m2H

0@ 1 + " �c�t�(1 + "=3)

�c�t�(1 + "=3) t2�(1 + �=3)

1A0@ c�

�s�

1A =

m2H

0@ c�(1 + ") + s�c�t�(1 + "=3)

�c�c�t�(1 + "=3)� s�t2�(1 + "=3)

1A = m2

h1

0@ c�

�s�

1A)

�1 + "+ t�c�t�(1 + "=3) = m2

h1=m2

H

c�t�(1 + "=3)=t� + t2�(1 + "=3) = m2h1=m2

H

)

t2�c�t�(1 + "=3) + t�(1 + "� t2�(1 + "=3))� c�t�(1 + "=3) = 0:

(7.324)

From this equation we can express the new angle � as an expression of the other angles � and � as follows

1� t2� =t�(1 + "� t2�(1 + "=3)

c�t�(1 + "=3))

t2� =2t�

1� t2�=

2c�t�(1 + "=3)

1 + "� t2�(1 + "=3)(7.325)

For �h = 0 (" = 0) we obtain

Page 184 of 193


tan 2� = cos �2 tan�

1� tan2 �= cos � tan 2�; (7.326)

which is the same result as below Eq. (24) in Ref. [3]. For small " (3�hv2 � m2

H) and s� we expand the

lowest mass eigenvalue as follows

m2h1 =

m2H

2

�1=c2� + "(1 + t2�=3)�

1

c2�

q(1 + "c2�(1 + t2�=3)� 4c4�t

2�(�c2�(1 + "=3)2 + (1 + ")(1 + "=3))

�

'm2H

2

h1=c2� + "(1 + t2�=3)� (1=c�)

q1 + 2"c2�(1 + t2�=3)� 4c4�t

2�(s

2� + 4"=3� 2c2�"=3)

i=m2H

2

�1=c2� + "(1 + t2�=3)� (1=c2�)

�1 + "c2�(1 + t2�=3)� 2c4�t

2�(s

2� + 4"=3� 2c2�"=3)

��

=m2H

�s2�s

2� + "s2�(4=3� 2c2�=3)

�:

(7.327)

For �h = 0 (" = 0) we obtain

m2h1 = m2

H sin2 � sin2 � (7.328)

which is the same mass as in Eq. (26) in Ref. [3]. Now, we will diagonalize the scalar mass matrix in

Eq. (7.313) and �nd the mass eigenstates. The eigenvalue equation and its mass solutions of this mass

matrix are

0 = det

0@ 1�X2 �t�

�t� t2� �X2

1A = (X2)2 �X2(1 + t2�))

X2 =1

2[1 + t2� �

q(1 + t�)2] =

1

2c2�(1� 1):

(7.329)

The masses of the charged pion states are

m2G� = 0 and m2

~�� = (m2H + �hv

2)=c2� : (7.330)

Thereafter, the mass eigenstates can be derived as follows0@ 1 �t��t� t2�

1A0@ A�

B�

1A =

0@ A� � t�B�

�t�A� + t2�B�

1A = 0 and

0@ 1 �t��t� t2�

1A0@ A+

B+

1A =

0@ A+ � t�B+

�t�A+ + t2�B+

1A =

1

c�

0@ A+

B+

1A ;

(7.331)

where the mass eigenstates are

G� = (s� ; c�)T = s��

�h + c��

� and ~�� = (�c� ; s�)T = �c��h + s��: (7.332)

The mass eigenstates of the neutral mass matrix in Eq. (7.318) have the same form as the charged mass

eigenstates, which can be written as

G3 = s��3h + c��

3 and ~�3 = �c��3h + s��3; (7.333)

Page 185 of 193


and the masses of these neutral pion states are the same as the charged pion states, i.e.

m2G3 = 0 and m2

~�3 = (m2H + �hv

2)=c2� : (7.334)

Couplings Normalized to the SM Couplings

In this subsection, the coupling of the light Higgs (h1) to the weak gauge bosons and the fermion Yukawa

coupling to the light Higgs (h1) h1 �ff are normalized to the couplings in SM. The Lagrangian terms with

these couplings in SM are

LSM =�SMfp

2h1ff + gSMhWWh1W

+� W

��; (7.335)

while the Lagrangian terms in Ref. [3] are

LUV =�PCHfp

2�hff + gPCH�WW�hW

+� W

�� +1

4c�s�fg

22�

4W+� W

��: (7.336)

where the second and third term come from the gauge-kinetic Lagrangian for the fundamental Higgs in

Eq. (5.7) and the �rst term in Eq. (5.24), respectively. We wish to found �V and �F in

�PCHf �hff =�F�SMf h1ff (7.337)�

gPCH�V V �h +1

2c�s�fg

22�

4

�W+� W

�� =�V gSMhV V h1W

+� W

��; (7.338)

which link the SM Lagrangian terms with the article Lagrangian terms. To do this we need to use that

v2EW = f2 sin2 � + v2 ,v2EWv2

=f2 sin2 �

v2+ 1 =

1

tan2 �+ 1 =

1

sin2 �;

(7.339)

and we know that

h1 = c��h � s��4;h2 = s��h + c��

4;(7.340)

therefore we have

�4 = �s�h1 + c�h2; (7.341)

�h = c�h1 + s�h2: (7.342)

The Yukawa coupling in the article can be expressed in terms of the Yukawa coupling as

�PCHf =mf

p2

v=mf

p2

vEW

vEWv

= �SMf1

sin�: (7.343)

By inserting this and the expression of �h in Eq. (7.342) into Eq. (7.337), we obtain

�PCHf �hff = �SMf1

sin��hff ! c�

s��SMf h1ff = �F�

SMf h1ff )

�F =c�s�:

(7.344)

Page 186 of 193


We know from SM that

gPCH�V V =g22v

2=g22vEW

2

v

vEW= gSMhV V sin�: (7.345)

By inserting this and the expression for �4 in Eq. (7.341) into Eq. (7.338), we obtain�gPCH�V V �h +

1

2c�s�fg

22�

4

�W+� W

�� !�gSMhV V s�c� �

1

2c�s�fg

22s�

�h1W

+� W

��

=

�s�c� � fs�

vEWc�s�

�gSMhV V h1W

+� W

�� = (s�c� � c�c�s�) gSMhV V h1W+� W

��

= �V gSMhV V h1W

+� W

�� ) �V = s�c� � c�c�s�;

(7.346)

where we have used

v2EW = f2 sin2 � + v2 ,v2EW

f2sin2�= 1 +

v2

f2 sin2 �= 1 + tan2 � =

1

cos2 �:

(7.347)

To summarize, we have obtained that

�F = c�=s� ;

�V = s�c� � c�c�s�:(7.348)

Page 187 of 193

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http://pdg.lbl.gov/2013/listings/rpp2013-list-w-boson.pdf

http://pdg.lbl.gov/2012/listings/rpp2012-list-rho-770.pdf

investigating a possible dynamical origin of the...

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