Examensarbete 15 hp Juni 2016

Unification in Particle Physics

Henrik Jansson

Kandidatprogrammet i Fysik Department of Physics and Astronomy Unification in Particle Physics

Henrik Jansson

A thesis presented for the degree of Bachelor of Science

Department of Physics and Astronomy Uppsala University, Sweden

June 2016 To my girls Unification in Particle Physics

Abstract

During the twentieth century, particle physics developed into a cornerstone of modern physics, culminating in the Standard Model. Even though this theory has proved to be of extraordinary power, it is still incomplete in several respects. It is our aim in this bachelor thesis to discuss some possible theories beyond the Standard Model, the main focus being on Grand Unified Theories, while also taking a look at attempts of further unification via discrete family symmetry. At the heart of all these theories lies the concept of local gauge invariance, which is introduced as a fundamental principle, followed by an overview of the Standard Model itself. No theory has so far managed to unify all elementary particles and their interactions, but some interesting features are highlighted. We also give a hint at some possible paths to go in the future in the quest for a unification in particle physics.

Sammanfattning

Under 1900-talet utvecklades partikelfysiken till en av de fundamentala teorierna inom fysiken, och kom att sammanfattas i den s.k. Standardmodellen. Aven¨ om denna modell r¨ont excep- tionella framg˚angervad g¨allerbeskrivningen av elementarpartiklar och deras v¨axelverkan, ¨ar den fortfarande ofullst¨andigp˚aflera s¨att.Syftet med denna kandidatuppsats ¨aratt diskutera m¨ojligateorier bortom Standardmodellen s˚asomStorf¨orenandeTeorier och diskreta famil- jesymmetrier vars avsikt ¨aratt koppla samman de tre familjerna av fermioner i Standard- modellen. Men f¨orstintroduceras id´enom lokal gaugeinvarians, vilken ligger till grund f¨or dessa teorier, varp˚aen ¨oversikt av Standardmodellen f¨oljer.Ingen teori har ¨annu lyckats ge en helt tillfredsst¨allandebild av elementarpartiklar och deras interaktion, men en del intressanta egenskaper hos f¨oreslagnateorier belyses i denna uppsats. Slutligen ges en del spekulativa f¨orslagp˚av¨ageratt g˚ai framtida f¨ors¨oktill f¨oreningarinom partikelfysiken. Acknowledgments

There are several people who deserve some special thanks in connection with the develop- ment of this thesis. First of all, I would like to sincerely thank my advisor Rikard Enberg for most generously sharing his time and deep knowledge of particle physics with me. Without his patience with all my questions this thesis would never have come to existence. I would also like to thank Gunnar Ingelman for accepting the role as subject reviewer, and Gabriella Andersson for being examiner of the thesis course. Both Gunnar and Gabriella also did a great job with a very appreciated study tour to the department of Physics and Astronomy at Uppsala University. Fredrik Gardell deserves some thanks for giving his opinion about a first draft of the thesis. Also, a big thank you to all my amazing students at T¨aby Enskilda gym- nasium who have been a great source of inspiration throughout the writing process. Finally, there are two people who have had an incredible patience with me during my work with this thesis, and to whom I am in great debt – Maria and Vendela. I hope that I some day can make this up for you.

The picture on the title page shows a three dimensional projection of a 24-cell. The ver- tices represent the root vectors of the Lie group Spin(8). The picture was generated by Robert Webb’s Stella software: http://www.software3d.com/Stella.php. Contents

1 Introduction 1 1.1 Background and Motivation ...... 1 1.2 Particle Physics as a Human Endeavor ...... 2 1.3 Method ...... 3 1.4 Outline and Contribution ...... 3 1.5 A Note on Units ...... 4

2 Background 5 2.1 Gauge Theories ...... 5 2.1.1 Local Gauge Invariance and QED ...... 5 2.1.2 Yang-Mills Theories ...... 7 2.1.3 The Strong Interaction – Quantum Chromodynamics ...... 9 2.2 The Standard Model ...... 9 2.2.1 The Weak and Electroweak Interactions ...... 10 2.2.2 Symmetry Breaking and the Higgs Mechanism ...... 11 2.2.3 Massive Fermions and Particle Generations ...... 12 2.2.4 Discrete Symmetries ...... 13

3 Theory 15 3.1 Grand Unified Theories ...... 15 3.1.1 The General Idea of GUTs ...... 15 3.1.2 A Prototypical Example ...... 16 3.1.3 SU(5), Spin(10) and the Pati-Salam Model ...... 19 3.2 Neutrino Oscillation and Discrete Flavor Symmetries ...... 22 3.3 A Note on SU(8) Grand Unification ...... 24

4 Discussion 25 4.1 Grand Unified Theories ...... 25 4.2 Discrete Flavor Symmetries ...... 28

5 Outlook 30 5.1 Substructure ...... 30 5.2 Unification of Quantum Numbers ...... 31 5.3 A Spin(8) Truly Unified Theory ...... 32

6 Conclusion 34 A Groups, Algebras, and Representations 36 A.1 Groups ...... 36 A.1.1 Finite and Discrete Groups ...... 37 A.1.2 Group Actions ...... 37 A.1.3 Direct and Semidirect Products of Groups ...... 38 A.1.4 Lie Groups ...... 38 A.2 Algebras ...... 39 A.2.1 Lie Algebras ...... 39 A.2.2 Clifford Algebras ...... 40 A.2.3 Tensor Algebras and Exterior Algebras ...... 41 A.3 Representations and Spinors ...... 42 A.3.1 Linear Representations ...... 42 A.3.2 Spinors ...... 44 A.4 Further Reading ...... 46

B Principal Fiber Bundles in Particle Physics 47 B.1 Differentiable Manifolds ...... 47 B.2 Derivatives, Differential Forms, and Pullbacks ...... 48 B.2.1 Left-invariant Vector Fields and the Cartan 1-form ...... 49 B.2.2 Killing Form and Riemannian Metric ...... 50 B.3 Principal Fiber Bundles ...... 50 B.4 Connections, Curvature, and Gauge Fields ...... 50 B.5 Matter Fields ...... 52 B.6 General Framework for Classical Gauge Theories ...... 53 B.7 Yang-Mills Theory ...... 54 B.7.1 Pure Yang-Mills Theory ...... 54 B.7.2 Pure Yang-Mills Electromagnetic Field Theory ...... 54 B.8 Further Reading ...... 55 Chapter 1

Introduction

1.1 Background and Motivation

Throughout the history of physics, unification of apparently different phenomena into a single framework has been a guiding light, and has shown to be of great importance and success. One famous example is Isaac Newton showing that the mechanical laws of the Earth and the heavens are the same. Another is the unification of electricity and magnetism into electro- magnetism described by Maxwell’s equations, which eventually led to Einstein’s unification of space and time into spacetime. During the course of the twentieth century, another area of physics saw some remarkable development, where unification from time to time was a neces- sity. This area is particle physics, which, in a modern sense, did not exist before the discovery of the electron at the end of the nineteenth century.1 During the first decades of the following century, particle physics was relatively simple – the only known particles were the electron, the proton, the neutron and the photon – the quantum of light. In the next couple of decades these particles got some new friends in terms of antiparticles such as the positron, and neutrinos (discovered experimentally in the mid- 1950s). During the 1950s, a whole lot of new particles were discovered in cosmic rays, which called for some unification in particle physics to bring some order in the chaos. A step in this direction was the Eightfold Way proposed by Gell-Mann (and independently by Ne’eman) in 1961. But this idea asked for a deeper explanation, which came in 1964 when Gell-Mann and Zweig invented the notion of quarks. In terms of quarks and antiquarks, the structure of all so called hadrons could be explained. However, these did not include the electron and the neutrino. Taking a step backward in time, inspired by some ideas in quantum mechanics, Heisenberg had tried to find a unifying theory for protons and neutrons, treating them as different states of the same particle. Even though his attempt was doomed to fail, it helped to develop the idea of local gauge invariance, which proved to be fruitful in the development of QED – quantum electrodynamics, a theory for the interaction of electrons and light. The local gauge symmetry in QED was given by the abelian group U(1). Later on, in the 1950s, Yang and Mills, inspired by the work of Heisenberg, generalized these ideas to non-abelian groups. This in turn was a necessity for the unification of electromagnetism with the action of the weak force, responsible for nuclear decay – the electroweak unification.

1The overview given in this section is primarily based on the introductory chapter in [1].

1 CHAPTER 1. INTRODUCTION 2

In the 1970s, another nuclear force, the strong force describing the interaction of quarks, was incorporated together with the electroweak force in a theory known as the Standard Model of particle physics. Its ambition was to describe all known particles and forces, except for gravity. The unification of the Standard Model and General Relativity – Einstein’s theory of gravity – is still to be worked out – if such a unification even exists. Although the Standard Model has been of exceptional success, it has several weaknesses, not only its lack of a treatment of gravity. Already in the 1970s there were suggestions how the Standard Model could possibly be improved. In this thesis we will take a look at some of these suggestions, with our main focus on Grand Unified Theories (or GUTs for short). How do they work and which role do they play in the quest for a unification of particles beyond the Standard Model? Are there other attempts of unification where Grand Unification also falls short? In order to unify particle physics with General Relativity – which is important for the understanding of cosmological phenomena – the Standard Model might have to be replaced by something else, and even though this “something else” is yet to be discovered, Grand Unification could be a hint of the right direction to go.

1.2 Particle Physics as a Human Endeavor

It is certainly a reasonable question to ask why people should bother doing particle physics at all, and why we should strive for unification. There are several possible answers to these questions. The first, and probably the most important reason, is curiosity. Throughout the entire history of mankind, people have always asked questions about the Universe and how it works. Particle physics seems to ask some of the deepest questions on the most fundamental level about the structure of the Universe as we know it today. Trying to answer these questions has taught us a great deal of science and of our own role in nature. Another reason for studying particle physics is the spin off effects on research and engi- neering. For example, the construction of the Large Hadron Collider and its experimental devices in CERN, Geneva, has occupied thousands of physicists and engineers for many years, and has pushed the limits of engineering and computer processing. This huge project has also brought together people from a large number of different countries and cultures, thus showing what human beings can achieve when collaborating. On the more theoretical side, particle physics has showed how quite abstract mathematical structures such as principal fiber bundles have a natural place in the description of the world (see [32]), and is an important example of how mathematics and physics have cross fertilized each other during the history of science. Theoretical physics is the biggest inspiration for, as well as the biggest consumer of, mathematics, thus leading to its further development. One example here is how Yang-Mills theories have played an important role in modern topology [30, 31]. The Royal Swedish Academy of Sciences has recently awarded two Nobel Prizes to discov- eries in the area of particle physics. The first was given to Higgs and Englert in 2013 for their theoretical prediction of the Higgs particle, which, through the Higgs mechanism (see chapter 2) gives all massive particles their mass. The Higgs particle was the last missing piece in the Standard Model. The second prize was awarded in 2015 to McDonald and Kajita for their contribution to the discovery of neutrino oscillation, which implies that neutrinos are massive (see chapter 3). This is not included in the Standard Model and shows a need for further CHAPTER 1. INTRODUCTION 3

unification in particle physics. Following the will of Alfred Nobel, these particle physicists “have conferred the greatest benefit to mankind.”

1.3 Method

The method for the thesis work has been literature studies and discussions with my advisor. Some important sources of information have been [1], [2] and [6] even though several other books and papers have been useful, such as [8] and [9].

1.4 Outline and Contribution

The thesis has the following structure. This introductory chapter contains a motivation for the choice of topic and what will be studied in the following chapters. Since the Standard Model and Grand Unified Theories are built on the notion of gauge theories, the introduction will be followed by a chapter about gauge symmetries as well as the Standard Model itself. Chapter 3 and 4 will be the heart of the thesis. In chapter 3 the concept of a Grand Unified Theory will be explained, first the general idea which will then be illuminated by some important examples. Also, some more recent attempts of cross generational unification with discrete groups will be introduced. Chapter 4 will discuss the strengths and weaknesses of the different theories in the previous chapter. After the main ideas have been covered, we will try to look at what could possibly be the future of unification in particle physics in chapter 5. This chapter will adress questions like “What is required of a theory to be a True Unification of all elementary particles and their interaction?” and will admittedly be of a more speculative nature than the rest of the text. A final chapter will wrap things up with a summary of the main ideas covered in the thesis. Also included are two appendices containing a brief overview of the mathematical foundations of particle physics. Except for some ideas in chapter 5, all the material is standard and can be found in the literature. Grand Unified Theories were developed already in the 1970s and no ground breaking work in this field has been done since then. In contrast, the research in family unification by discrete groups is a more active area still in progress and we will try to give an overview of the current status of the subject. The ambition from the beginning was to develop some of the speculative ideas now put in chapter 5, but after a second thought this seemed to be an unreasonable goal for a bachelor thesis – both because the mathematical machinery involved would be too heavy, and because the whole idea could easily lead to a dead end. Also, a first thought was to formulate the whole thesis using the Cartan formalism and the language of principal fiber bundles, but since this is not at an undergraduate level, it was dispatched to an appendix. Hopefully this appendix can be to some use for a graduate physics student who wants to learn more about the formal mathematical language of particle physics. Therefore, I claim no novelty in the ideas in this thesis – my contribution is instead to give a coherent and readable introduction to unification in particle physics available to an undergraduate physics student with an interest in the subject. CHAPTER 1. INTRODUCTION 4

1.5 A Note on Units

It is customary in high energy physics to work in natural units, i.e. units where ~ = c = 1. We will stick to this convention, even though one of the main references for chapter 2, [1], does not. Although the physical content of several equations and formulas might not be as explicit as when ~ and c are kept, natural units will give a cleaner exposition. Also, in the context of electromagnetism, we will set 0 = µ0 = 1. Chapter 2

Background

2.1 Gauge Theories

One of the most important ideas in twentieth century physics was the principle of local gauge invariance. Not only was it of fundamental importance to the unification of particle physics in the theories leading to the Standard Model and Grand Unified Theories, but General Relativity can also be considered as a gauge theory. It was actually in this later context it had its origin, both the concept and the name, as Hermann Weyl in 1918 tried to unify General Relativity with electromagnetism [8]. His idea was to allow the rescaling of physical objects at each individual point in spacetime, i.e. change their “gauge” locally. The price he had to pay was the introduction of a “connection field” which tells how the scale of length has to be adjusted from point to point. The remarkable thing was that the connection field satisfied Maxwell’s equations. However, since there are objective sizes of e.g. elementary particles which do not change from one point to another, Weyl’s local gauge theory had at that time to be abandoned. But at the end of the 1920s a new theory, called quantum electrodynamics, came to incorporate Weyl’s idea in a crucial way.

2.1.1 Local Gauge Invariance and QED Quantum Electrodynamics (or QED for short) is the quantum field theory of the interaction of photons (quanta of the electromagnetic field) with electrons and positrons. It is the simplest and also most precise quantum field theory which exists. The concept of quantum field theory – which unifies quantum mechanics with special relativity – is deep and not yet fully understood, and is beyond the scope of this thesis, but we will do our best to understand QED in terms of classical field theory. A formulation of classical mechanics is built on the so called Lagrangian function,

L = T − U (2.1) where T is the kinetic energy and U is the potential energy in a system [1]. The Lagrangian L is a function of the coordinates qi and their time derivativesq ˙i, i = 1, . . . , n. Hamilton’s variational principle gives the following equations of motion [10]:

d  ∂L  ∂L = (2.2) dt ∂q˙i ∂qi

5 CHAPTER 2. BACKGROUND 6

These so called Euler-Lagrange equations are suitable for a localized entity such as a particle in the classical sense. However, when dealing with relativistic fields, e.g. the electromagnetic field, the coordinates qi have to be replaced by fields which are functions of position in µ 0 1 2 3 spacetime, i.e. φi(t, x, y, z) = φ(x ) where x = t, x = x, x = y and x = z, and ∂φ ∂ φ ≡ i (2.3) µ i ∂xµ

The Lagrangian will be replaced by a Lagrangian density L which will be a function of φi and ∂µφi. (With a slight abuse of language, it is also often called the Lagrangian.) The Euler-Lagrange equations will take the form

 ∂L  ∂L ∂µ = (2.4) ∂(∂µφi) ∂φi Here the Einstein summation convention is implicit on the left hand side. In the Standard Model there are three different kinds of fields. The first is a scalar (spin-0) field, φ, (the Higgs field) having Lagrangian

1 µ 1 2 2 L0 = (∂µφ)(∂ φ) − m φ (2.5) 2 2 The second kind is a vector (spin-1) field, Aµ, which has the Lagrangian

1 µν 1 2 ν L1 = − F Fµν + m A Aν (2.6) 4 2 where F µν ≡ ∂µAν − ∂νAµ. The field equation becomes

µν 2 ν ∂µF + m A = 0 (2.7) which for a massless field (m = 0) gives Maxwell’s equations for empty space [1]. (Here the “vector potential” Aµ is considered the fundamental quantity rather than the field F µν.) Finally, the third kind of field in the Standard Model is a spinor field, ψ, with spin 1/2 (the leptons). The Lagrangian of this kind of field is

¯ µ ¯ L1/2 = iψγ ∂µψ − mψψ (2.8) where γµ are the Dirac gamma matrices (see Appendix A). The Euler-Lagrange equations will in this case give the Dirac equation for a particle and its antiparticle of mass m. Now ψ is a wave function of a particle, and is only determined up to a phase θ, i.e. eiθψ, where θ is any real number, will represent the same physical state of the particle. Also, the Lagrangian L1/2 is invariant under such a global phase transformation. However, making the transformation local, i.e. ψ → eiθ(x)ψ, where x = xµ is a spacetime coordinate, will destroy the invariance: ¯ µ L1/2 → L1/2 − (∂µθ)ψγ ψ (2.9) 1 or, if λ(x) ≡ − q θ(x), where q should be thought of as the charge of the particle, ¯ µ L1/2 → L1/2 + (qψγ ψ)∂µλ (2.10) CHAPTER 2. BACKGROUND 7

But the invariance of the Lagrangian under local phase transformations can be reinforced if we are willing to pay the price of introducing a gauge field Aµ into the Lagrangian:

¯ µ ¯ ¯ µ L1/2 = (iψγ ∂µψ − mψψ) − (qψγ ψ)Aµ (2.11)

The gauge field changes according to the gauge transformation

Aµ → Aµ + ∂µλ (2.12)

The new field also comes with a self interaction in terms of the Lagrangian

1 µν 1 2 ν L = − F Fµν + m A Aν (2.13) 4 2 A ν While the first term is invariant under a gauge transformation, A Aν is not, implying the requirement of a massless gauge field: mA = 0. Thus, taking a spin 1/2 field under the consideration of local gauge invariance, results in the Lagrangian of quantum electrodynamics

¯ µ ¯ 1 µν ¯ µ L = (iψγ ∂µψ − mψψ) − F Fµν − (qψγ ψ)Aµ (2.14) 4

where Aµ is a massless spin 1 gauge field (a photon) interacting with the spinor field (elec- tron/positron) through the last term. The extra term which is introduced in the Langrangian when the global change of phase is made local originates from the differentiation – in terms of the function λ = λ(x) the transformation reads −iqλ ∂µψ → e (∂µ − iq∂µλ)ψ (2.15) By defining a differential operator, called the covariant derivative, as

Dµ ≡ ∂µ + iqAµ (2.16) a gauge transformation of Aµ will imply the transformation

−iqλ Dµψ → e Dµψ (2.17)

Using the covariant derivate instead of the ordinary derivative in a Lagrangian will turn its global invariance into a local one [1].

2.1.2 Yang-Mills Theories

In QED the change of phase is given by multiplying the wave function by a factor eiθ(x), which is an element of the Lie group U(1). We therefore say that QED is a gauge theory with gauge group U(1). (For an introduction to (Lie) group theory, see Appendix A. Also, Appendix B contains the formulation of gauge theories in terms of so called principal fiber bundles – the natural setting for a mathematical formulation of particle physics.) In quantum mechanics, the different spin states of a spin 1/2 particle (e.g. an electron) is described by a two dimensional representation (a so called doublet) of the non-abelian Lie group SU(2) [11]. (For the notion of a representation, see Appendix A.) Inspired by this mathematical formalism, Heisenberg made an early attempt of unification in particle physics – using mass as an approximate symmetry he tried to identify the proton and the neutron as CHAPTER 2. BACKGROUND 8

two different states of the same particle, placing them in the same doublet of SU(2) [2]. Even though this construction broke down, it gave Yang and Mills the idea in 1954 of formulating a non-abelian gauge theory based on the group SU(2). Yang-Mills theories with gauge groups SU(2) and SU(3) are at the heart of the Standard Model, giving rise to the weak and the strong interaction, respectively, and also give the structure of GUTs. Thus it will be worthwhile to spend some time and energy to see how they work. The main focus will be on the SU(2) theory, but an overview of how it generalizes to the SU(3) case will also be given. In QED the gauge transformation manifests itself in terms of a multiplication by a complex number of modulus 1. In an SU(2) theory, such numbers will be substituted by unitary 2 × 2 matrices U, which can be written as

U = eiθeiτ·a (2.18) where τ ·a = τ1a1 +τ2a2 +τ3a3 and τi are the Pauli matrices. For a global SU(2) symmetry, the ai are numbers, whereas for a local SU(2) gauge symmetry, these are functions of spacetime: a = a(x). Now there will be two spin 1/2 fields put into a column vector

 ψ  ψ ≡ 1 (2.19) ψ2

and the Lagrangian is ¯ µ ¯ L = iψγ ∂µψ − mψψ (2.20) where m is the mass of each of the two particles. To make the Lagrangian invariant under the local SU(2) transformation ψ → eiτ·a(x)ψ (2.21) we use the recipe of making the derivative covariant:

Dµ ≡ ∂µ + iq(τ · Aµ) (2.22)

where in this case there are three (massless) gauge fields – one for each dimension of SU(2) – given by the components of Aµ = (A1µ,A2µ,A3µ). In the total Lagrangian, self interaction terms of these fields have to be included; the sum of these is

1 µν LA = − F · Fµν (2.23) 4 where this time F µν ≡ ∂µAν − ∂νAµ − 2q(Aµ × Aν) (2.24) This will result in the total Lagrangian

¯ µ ¯ 1 µν ¯ µ L = (iψγ ∂µψ − mψψ) − F · Fµν − (qψγ τψ) · Aµ (2.25) 4 CHAPTER 2. BACKGROUND 9

2.1.3 The Strong Interaction – Quantum Chromodynamics It was first with the use of the Higgs mechanism (see section 2.2) that an SU(2) Yang-Mills theory could give rise to massive gauge fields in the weak interaction. The first Yang-Mills theory with massless gauge fields, so called gluons, concerned the strong interaction of quarks. This theory – Quantum Chromodynamics (QCD) – emerges from a local SU(3) symmetry, where the role of electric charge in QED is played by three new quantum numbers, called col- ors. These numbers, red, green and blue, were introduced in order to save the Pauli exclusion principle [1]. Even though these colors of quarks have nothing to do whatsoever with color of light, they offer a nice mnemonic for a rule saying that free particles are colorless – adding an equal amount of red, green and blue light results in white. In the language of representation theory, it means that free particles are singlets (i.e. one dimensional representations) of the gauge group SU(3). When combining three quarks into a baryon, all three quarks have to be of different color, while making a quark-antiquark pair into a meson, the antiquark need the anticolor of the quark. (The whole whimsical color business could preferably be avoided by instead using the names x, y and z for the new quantum numbers and give them a geometrical interpretation – see section 5.2.) The Lagrangian formalism of QCD will be very similar to that of the SU(2) Yang-Mills theory sketched above. The original free Lagrangian will still look the same as in (2.20) even though in this case   ψr ψ ≡  ψg  (2.26) ψb

There will be eight massless gauge fields Aµ = (A1µ,...,A8µ) (one for each of the dimensions of SU(3)) and the Pauli matrices will be replaced by the eight Gell-Mann matrices λ = (λ1, . . . , λ8). Everything will look the same as in (2.21) - (2.25) if τ is substituted by λ and the cross product in (2.24) is substituted by the Lie bracket in su(3), the Lie algebra of SU(3). (See Appendix A for a treatment of Lie algebras.)

2.2 The Standard Model

In the mid-1970s, around the time of the discovery of the charm and bottom quarks, all known elementary particles and their strong, weak and electromagnetic interaction were united in the Standard Model of particle physics. The Standard Model also predicted the existence of several particles which at that time had not yet been observed, namely the top quark, the tau neutrino, the weak vector bosons and the Higgs boson (see below). The later discovery of these particles only confirmed the incredible success of the theory seen in earlier experiments and observations. At the same time, there are several hints that the Standard Model is not the final solution to the deep mysteries of particle physics. But before we address some of these issues and what might be their solution in the next chapter, we will in this section give an overview of the Standard Model itself. First of all, there are two groups of players on the arena of elementary particle physics – fermions and bosons (named after Enrico Fermi and Satyendra Nath Bose, respectively). The fermions all have spin 1/2, and are of two types: leptons and quarks. The leptons are the electron (e−) and its heavier unstable cousins, the muon (µ−) and the tau (τ −), together with one neutrino of each kind: νe, νµ and ντ . In the Standard Model neutrinos are assumed CHAPTER 2. BACKGROUND 10

Table 2.1: Elementary fermions in the Standard Model

Name Symbol Electric charge Color charge electron e− −1 0 muon µ− −1 0 tau τ − −1 0 electron neutrino νe 0 0 mu neutrino νµ 0 0 tau neutrino ντ 0 0 2 up quark u 3 r/g/b 1 down quark d − 3 r/g/b 1 strange quark s − 3 r/g/b 2 charm quark c 3 r/g/b 1 bottom quark b − 3 r/g/b 2 top quark t 3 r/g/b massless (for no good reason [1]), even though later experiments have shown that they in fact have a small but nonzero mass (see the next chapter). So there are six leptons in total. The number of quarks is the same, and they come in the following so called flavors: up (u), down (d), strange (s), charm (c), bottom (b), and top (t). Each of the twelve fermions also has a partner with the same mass but with the opposite electric charge – its antiparticle. Antiparticles are usually denoted by an overbar, e.g.u ¯ is an anti-up quark andν ¯e is an anti-electron neutrino. However, the antiparticles of the charged leptons are denoted by a plus sign, e.g. e+ is a positron, the antielectron. An overview of the fermions are listed in Table 2.1. Now to the bosons. These are also of two different types: the vector bosons and the scalar Higgs boson. The vector bosons are also called gauge bosons where the “gauge” prefix has its origin in the fact that these bosons arise as gauge fields when local gauge invariance is demanded of the Lagrangian of a fermion field as explained in section 2.1. There are nine massless vector bosons: the photon (γ) coming from the U(1) symmetry of electromagnetic interaction, and eight different gluons (all denoted by g) arising from the SU(3) symmetry of the strong interaction. However, there is a third symmetry involved in the Standard Model, given by the weak interaction.

2.2.1 The Weak and Electroweak Interactions While the QED and QCD parts of the Standard Model are relatively straight forward once the idea of local gauge invariance is accepted (and quantum field theory is ignored), the third gauge symmetry – the weak SU(2) symmetry – is somewhat more involved. Let us take a look at some of its special properties. The weak interaction was first introduced in an attempt to explain radioactive decay [2]. (It was also in that context neutrinos were invented in order to restore conservation of energy in β decay.) A fact, discovered in the 1950s, which is crucial to the structure of the Standard Model is that the weak interaction only involves so called left-handed particles (and right- handed antiparticles). The “handedness” (or helicity) of a particle is determined by its spin CHAPTER 2. BACKGROUND 11 in its direction of motion, and is either “left” (L) or “right” (R). The quantum numbers associated with electromagnetism and the strong interaction are electric charge and color, respectively. The weak interaction comes with a quantum number called (weak) isospin, which is not to be confused with the “strong isospin” of Heisenberg’s attempt to unify the nucleons. 1 The third component of isospin is denoted by I3 and is ± 2 for left-handed fermions, while it − 1 − vanishes for their right-handed companions. For example, I3(eL ) = − 2 but I3(eR) = 0. In Heisenberg’s strong isospin theory, the proton and the neutron were put in the same fundamental two dimensional representation of the group SU(2), i.e. making a doublet. In the (weak) isospin theory of the Standard Model, the doublets are inhabited by left-handed fermions teamed up in pairs, e.g. one lepton and one quark pair would be     νeL uL − and (2.27) eL dL − The corresponding right-handed particles, i.e. νeR, eR, uR and dR, are singlets under SU(2). In the Standard Model there is yet another quantum number, called hypercharge and is denoted by Y . Hypercharge is used to label representations of SU(2) and its relation to electric charge Q and isospin I3 is given by the Gell-Mann–Nishijima formula [2]: Y Q = I + (2.28) 3 2 To get an idea of the meaning of hypercharge, we first have to take a further look at the isospin SU(2) symmetry. As was noted in section 2.1, such a symmetry will give rise to three gauge fields through the principle of local gauge invariance. The particles associated to these fields are known as weak vector bosons and are denoted by Z0, W + and W −. The boson associated with the hypercharge is nothing but an old and familiar friend – the photon. A crucial part of the Standard Model is the unification of the electromagnetic and weak interactions into the electroweak interaction, first proposed by Weinberg and Salam. In this unification the weak bosons will be massive, while the photon remains massless. The electroweak theory is determined by a U(1)×SU(2) symmetry. This product group is no longer a simple Lie group, and this has consequences for the structure of the theory (see the next section and chapter 4). In the unification there will be four massless gauge fields, one for each of the dimensions of U(1)×SU(2). As was mentioned above, the bosons for three of them – the weak bosons corresponding to the SU(2) part – are massive. How can that be accomplished?

2.2.2 Symmetry Breaking and the Higgs Mechanism Let us take a look at the Lagrangian formalism for the electroweak unification and how some gauge fields can gain mass. We start with the Lagrangian

† µ † L = ∂µΦ ∂ Φ − V (Φ Φ) (2.29) which has a global U(1)×SU(2) symmetry [3]; here Φ has two complex component fields  Φ  Φ = A (2.30) ΦB and Φ† denotes the hermitian conjugate. To make the global symmetry local, we should make the derivative covariant; in this case four fields will contribute to Dµ. One of them, CHAPTER 2. BACKGROUND 12

denoted by Bµ(x) = Bµ(x)τ0, where τ0 is the 2 × 2 identity matrix, is given by the U(1) symmetry. The remaining three, arising from the SU(2) symmetry, can be summarized in P3 k Wµ(x) = k=1 Wµ (x)τk. This gives the covariant derivative

ig1 ig2 Dµ = ∂µ + Bµ + Wµ (2.31) 2 2 where g1 and g2 are so called coupling constants of the electromagnetic and weak interaction, respectively. The locally gauge invariant Lagrangian thus takes the form

† µ † L = (DµΦ) D Φ − V (Φ Φ) (2.32)

In order to make the Wµ fields massive, the SU(2) symmetry has to be broken. This can be accomplished in the following way. Let

2 † m † 2 2 V (Φ Φ) = 2 (Φ Φ − φ0) (2.33) 2φ0 where φ0 is a real parameter. V as a function of |Φ| will have a minimum on the three 2 2 2 2 dimensional sphere |Φ| = |ΦA| + |ΦB| = φ0 in four dimensional (real) space. Making a local SU(2) gauge transformation keeps Φ on this sphere. By choosing a specific gauge where ΦA = 0 and ΦB = φ0 is real, we break this symmetry and get a specific ground state     ΦA 0 Φground = = (2.34) ΦB φ0

corresponding to the energy of the√ vacuum. In excited states of the ground state φ0 will gain an additional term: φ0 + h(x)/ 2 where h(x) is a real field. A quantum of the field h(x) is called a Higgs boson; such a particle of mass 125 GeV was found in CERN in 2012. The bosons of the Standard Model can now be summarized in Table 2.2.

Table 2.2: Elementary bosons in the Standard Model

Name Symbol Electric charge Color charge Mass (GeV) photon γ 0 0 0 neutral weak boson Z0 0 0 91.2 charged weak bosons W ± ±1 0 80.4 gluons g 0 r/g/b 0 Higgs boson H0 0 0 125

2.2.3 Massive Fermions and Particle Generations Some vital parts of the Standard Model have not yet been mentioned. One of these is how the charged leptons and quarks gain their mass. This is done by adding gauge invariant terms to the Lagrangian where fermion fields are coupled to the Higgs field Φ (so called Yukawa coupling). E.g. for an electron we get, after symmetry breaking, in terms of Weyl spinors, the contribution

e † † ceh  † †  L = −ceφ0(e eR + e eL) − √ e eR + e eL (2.35) mass L R 2 L R CHAPTER 2. BACKGROUND 13 where ce is a dimensionless√ coupling constant related to the electron mass me and the Higgs − − field φ0 by ce = me/ 2φ0 [3]. Similar terms will be added for the µ and the τ with coupling constants cµ and cτ , respectively. For the quarks the story goes more or less the same way, but in that case left-handed quarks are put in SU(2) doublets, making the expressions in the Lagrangian a bit more complicated. (No such doublets showed up for the leptons since neutrinos are massless in the Standard Model.) Another feature of the Standard Model which has not been mentioned so far, but which plays a prominent role in its structure, is the existence of fermion generations or families. (We will use these words interchangeably even though their use in everyday language suggests otherwise.) When the quark model was first introduced, it only postulated two different flavors of quarks, up and down. Together with the electron and its neutrino, these particles build up the first generation – their left handed versions are put in doublets of SU(2) as shown in (2.27). Ordinary matter is completely made from first generation fermions – the quark content of the proton and the neutron is uud and udd, respectively. When more and more “strange particles” showed up in the mid-twentieth century, Gell- Mann used the Eightfold Way pattern built on an approximate SU(3) mass symmetry to bring order in the chaos. By adding a third quark flavor, the strange (s) quark, he could give the Eightfold Way a more sophisticated appearance. It also allowed him to predict the existence of a new particle – the Ω− with quark content sss. At the same time there were two additional leptons, namely the muon and its neutrino. In order to get a complete second generation of fermions, a fourth quark flavor, charm (c), was suggested. It later turned out to be useful in the description of new hadrons. (Since there were now two generations of fermions, the Eightfold Way was in a sense the first attempt of a cross generational unification in particle physics, even though it was proposed before the idea of fermion families had been introduced.) In addition, there is a third generation which repeats the pattern of the first two. It contains the tau lepton and its neutrino, and two more quark flavors, bottom (b) and top (t). The generational pattern is summarized in Table 2.3.

Table 2.3: Fermion generations in the Standard Model

Generation: First Second Third Quarks u c t d s b Leptons e− µ− τ − νe νµ ντ

2.2.4 Discrete Symmetries The three symmetries of the Standard Model considered so far are continuous, i.e. given by Lie groups – U(1), SU(2), and SU(3). However, there are also three discrete symmetries involved, called charge conjugation (C), parity (P) and time reversal (T). These are all given by the two element group Z2, consisting of the identity element and an involution, denoted by C, P , and T , respectively. The parity operator P changes the direction of all three spatial coordinates, i.e. P (x, y, z) = (−x, −y, −z). Before the 1950s, it was believed that all physical processes are invariant under CHAPTER 2. BACKGROUND 14 parity symmetry, but surprisingly an experiment proposed by Lee and Yang, and performed by Wu, in 1956, showed that the weak interaction is violated by parity transformations. Since neutrinos are electrically neutral and colorless, they only interact weakly. This has as a con- sequence that all neutrinos in the Standard Model are left-handed, while all antineutrinos are right-handed. Despite its name, the charge conjugation operator C does not only change the charge of a particle, but also its baryon number, lepton number, and quark numbers. Written as a multiplicative quantum number, charge conjugation is conserved in the electromagnetic and strong interactions, but not in the weak interaction, as can be seen by considering neutrinos: all neutrinos are left-handed, but there are no left-handed antineutrinos. (C leaves spin and momentum unchanged, and so the helicity remains the same.) While parity and charge conjugation are not conserved individually by the weak interac- tion, their combination – CP symmetry – is conserved. That is, in most cases. Experiments with neutral kaons done by Cronin and Fitch in 1964 showed CP violation [1]. But then there is the third discrete symmetry, time reversal, which runs all processes backwards. A result in quantum field theory, known as the TCP theorem, says that the combination of all three discrete symmetries (in any order) is an exakt symmetry of all interactions, given some reasonable assumptions of the theory, such as Lorentz invariance. In the case of the neutral kaon CP violation, this means that there must be a compensating violation of T as well. Chapter 3

Theory

3.1 Grand Unified Theories

In this chapter we will take a look at unification in particle physics beyond the Standard Model, starting in this section with Grand Unified Theories (GUTs). In the next section family unification with discrete flavor groups will be covered. But before digging into these further unifications, we should first adress a question important to ask at this stage: Why is the Standard Model not a satisfactory theory of elementary particles and their interaction? After all, it has made some marvelous predictions, and has stand the test of most experiments and observations for the last forty years. At a first stage, the reasons are two-fold. On the one hand, there are several physical phenomena which the Standard Model fails to explain, e.g. the existence of dark matter and massive neutrinos. On the other hand, the theory has some structural problems. One of them is the fact that there are too many free parameters which have to be deduced from experiments rather than being explained by theory, e.g. the different masses of the elementary particles, and the gauge and Yukawa couplings [4]. Furthermore, there is no explanation of the generational pattern of fermions or the quantization of electric charge. Also, from a mathematical point of view the Standard Model is a big mess, even though it is built on fundamental principles as Hamilton’s variational principle and the principle of local gauge invariance. It gives the impression of a patchwork, which every now and then has been completed with another matching piece of information such as the discovery of a new particle or the theoretical assumption of a new kind of symmetry. This might be natural from a historical point of view, but is nevertheless no excuse for a lack of deeper explanation of fundamental concepts. Making a comparison to General Relativity – a theory the Standard Model refuses to be incorporated with – it is hopelessly behind in clarity and simplicity. To make a marriage of particle physics with gravity a possibility, the Standard Model will probably have to leave room for something more profound and elegant. Grand Unification might point in the right direction to go and look for this Holy Grail of particle physics.

3.1.1 The General Idea of GUTs The Standard Model does not only contain the electroweak unification but also Quantum Chromodynamics, resulting in the (non-simple) gauge group

GSM = U(1) × SU(2) × SU(3) (3.1)

15 CHAPTER 3. THEORY 16

The covariant derivative leading to local gauge invariance of the Lagrangian will now take the form ig1 ig2 ig3 Dµ = ∂µ + Bµ + Wµ + Gµ (3.2) 2 2 2 P8 k where g3 is the coupling constant of the strong interaction, and Gµ(x) = k=1 Gµ(x)λk. This is what it looks like in the low energy domain of the present Universe. The whole idea of Grand Unification is to replace the Standard Model gauge group GSM with a simple group GGUT – one of the benefits of this is that there will be a single coupling constant g determining the strength of the coupling between the gauge bosons and the fermions. In Grand Unification there will be one gauge boson for each of the dimensions of GGUT, i.e. the total bosonic field Pn k is Wµ(x) = k=1 Wµ (x)ek where n = dim GGUT and ek are generators of the Lie algebra gGUT of GGUT. The covariant derivative will now return to the simple form ig Dµ = ∂µ + Wµ (3.3) 2 Thus, in a GUT the three different interactions – electromagnetic, weak, and strong – are unified into a single interaction (gravitation is still missing). But this symmetry – if it exists at all – is rather well disguised at low energies. It is only on a very high energy scale, somewhere at the order of 1015 GeV (the so called GUT scale), where Grand Unification is supposed to have been realized in the early Universe. To get from there to here, i.e. from Grand Unification to the Standard Model, some symmetry breaking must have been taken place. How this is realized is dependent on the choice of GUT group, and for a specific choice there might still be different paths to go. But one or several Higgs fields have to be introduced to achieve the breaking of the GUT symmetry. Let us take a look at an example of Grand Unification to get a better feeling for the concepts involved.

3.1.2 A Prototypical Example The first example of a GUT was given by Georgi and Glashow in 1974, about the same time as the Standard Model was completed. Thus it was realized already at the time of its invention that the Standard Model left some important questions unanswered. This first GUT was built on the gauge group SU(5) and often serves as a prototypical example. To find the order of the GUT scale in this case, MSU(5), the running of the couplings have to be applied: when the so called energy scale of renormalization, µ, changes, so do the strengths of the couplings, 1 αi, i = 1, 2, 3. It is customary to consider the inverses of the couplings, which are given by M  α−1(µ) = α−1(M ) − 2b log SU(5) (3.4) i i SU(5) i µ where 1  4  b = 11 − N (3.5) 3 4π 3 f 1 22 4 1  b = − N − N (3.6) 2 4π 3 3 f 6 H 1 4 1  b = − N + N (3.7) 1 4π 3 f 10 H

1The idea of renormalization is beyond the scope of this thesis, even though it is crucial in order to get testable results from quantum field theoretical calculations. It would be too technical and take us too far aside to dig into these matters here. The interested reader could consider [5] for an introduction. CHAPTER 3. THEORY 17

are so called one-loop beta function coefficients [4]. Here Nf is the number of fermion fami- lies, and NH is the number of Higgs doublets. Figure 3.1 shows the running of the couplings, assuming Nf = 3 and NH = 1. As is clear from the picture, the couplings do not meet at a single point, giving us a hint that there is no Grand SU(5) Unification after all – possi- ble remedies will be discussed in the next chapter. Still we can conclude that there is an approximate unification at around 1015 GeV.

Figure 3.1: Running of the (inverted) coupling constants. Note the logarithmic scale on the energy axis (GeV). Here the solid line shows the electromagnetic coupling, the dashed line the weak coupling and the dotted line the strong coupling.

In the Standard Model the different elementary particles are accomodated in different representations of the groups U(1), SU(2) and SU(3), respectively. Gauge bosons are placed in the adjoint representations, i.e. in the Lie algebras u(1)⊕su(2) (the four electroweak bosons) and su(3) (eight gluons interacting strongly). Left handed fermions are put in doublets of SU(2), where quarks are triplets and leptons are singlets of SU(3). Finally, the Higgs boson is a SU(2) doublet. Let us see how particles are distributed among different irreducible representations of the Grand Unification group SU(5).2 To begin with, the gauge bosons are once again put in the adjoint representation su(5), which is spanned by La, a = 1,..., 24. The eight gluons are included by inserting the Gell- Mann matrices in the upper left 3 × 3 block:

 0 0   λa 0 0    La =  0 0  (3.8)    0 0 0 0 0  0 0 0 0 0 for a = 1,..., 8. The charged weak bosons are taken care of by the non-diagonal Pauli

2Here we follow [6]. CHAPTER 3. THEORY 18 matrices in the lower right 2 × 2 block:  0 0 0 0 0   0 0 0 0 0    L9,10 =  0 0 0 0 0  (3.9)   0 0 0  τ  0 0 0 1,2 Since the rank of SU(5) is 4, there are two more diagonal generators except for the two gluons L3 and L8. These will correspond to the two neutral electroweak bosons, i.e. the photon and Z0, and are given the matrix representations

L11 = Diagonal(0, 0, 0, 1, −1) (3.10) and 1 L12 = √ · Diagonal(−2, −2, −2, 3, 3) (3.11) 15 where the normalization is chosen such that Tr(LaLb) = 2δab. Note that L12 is traceless as is demanded of elements of su(5), even though neither the upper 3 × 3 nor the lower 2 × 2 diagonal submatrix is. So much for the known gauge bosons. However, the adjoint representation of SU(5) leaves room for another twelve bosons, L13,...,L24. These come in six particle-antiparticle pairs, denoted by Xi/X¯i and Yi/Y¯i, i = 1,..., 6, and correspond to generators of the off block diagonal entries in su(5). One of the peculiarities of these new bosons is that they can couple leptons to quarks, and are therefore sometimes called “leptoquark” bosons. See Chapter 4 for a discussion of some of the consequences of this interaction. Let us now turn to fermion representations. One right handed quark of three different colors is, together with the right handed anti-leptons of the same generation, put in a funda- mental five dimensional representation, e.g.  dr   dg    5 =  db  (3.12)    e+  ν¯ e R (A representation is often denoted by a boldface number indicating its dimension.) Observe that the Standard Model only contains left handed neutrinos, which are equivalent to right handed anti-neutrinos – such a neutrino is put in the 5 above. We can write 5 = (3, 1) ⊕ (1, 2) under SU(3)×SU(2) (3.13) to indicate that the quark part of 5 is a triplet and the lepton part is a singlet under SU(3), while the quark part is a singlet and the lepton part is a doublet under SU(2). The leptoquark bosons will however act nontrivially on both the quarks and the leptons, making an interaction between these two categories of particles possible. The next step is to find a representation which can hold the remaining fermions in the first family: the left handed positron and down quark, and the left and right handed up quark. One way to build new representations from old ones is to take their tensor product, e.g. 5 ⊗ 5 = 10 ⊕ 15 (3.14) CHAPTER 3. THEORY 19

The ten dimensional part can be represented by an antisymmetric matrix containing exactly the missing particles:

 r r  0u ¯b −u¯g −u −d g g  −u¯b 0 −u¯r −u −d   b b  10 =  u¯g −u¯b 0 −u −d  (3.15)    ur ug ub 0 −e+  dr dg db e+ 0 L The total fermion representation for the first family is 5¯ ⊕ 10; here 5¯ is the conjugate repre- sentation of 5 – it is used to get the same chirality of all the particles in the direct sum. Just as in the Standard Model there is one such direct sum for each family [2]. Now there is one last piece in the particle puzzle, namely the symmetry breaking Higgs bosons. In the Standard Model, a Higgs particle is required to break GSM = U(1)×SU(2)× SU(3) down to U(1)×SU(3), thereby giving the SU(2) weak bosons mass. In a GUT there must be at least one additional Higgs boson in order to break GGUT (in this case SU(5)) down to GSM. In this step the X and Y bosons will gain mass of the order of MSU(5), while the Standard Model bosons remain massless. This extra Higgs boson is given by the 24 dimensional adjoint representation 24H = su(5) of real scalars. Its vacuum expectation value (vev) is given by 3 3 h24 i = v · Diagonal(1, 1, 1, − , − ) (3.16) H 2 2

Since h24H i is proportional to the identity matrix in both the SU(3) and SU(2) parts, only the X and Y bosons will gain mass. The Higgs field of the Standard Model, Hf , is put in a five dimensional fundamental representation   Hc 5H = (3.17) Hf together with the color triplet Hc. 5H gives mass to the fermions through Yukawa coupling. The mass of the colored part Hc must be at the MSU(5) scale, or it will give rise to proton decay [4]. (For this and other issues with GUTs, see the next chapter).

3.1.3 SU(5), Spin(10) and the Pati-Salam Model Experiments have shown that neutrinos are in fact massive. (This is implied by so called neutrino oscillation – see section 3.2). As a consequence there must be right handed neutrinos, a fact which is not accomodated for in the Standard Model. It would be nice if a Grand Unified Theory had a natural mechanism for right handed neutrinos. In the SU(5) GUT this is not the case – the right handed neutrino has to be added in an ad hoc manner in a one dimensional representation: 16 = 1 ⊕ 5¯ ⊕ 10 (3.18) A GUT which does come equipped with a right handed neutrino is based on the gauge group Spin(10). Often this group is referred to as SO(10) in the GUT literature, but this is rather misleading. Spin(10) is the double cover of SO(10) and has certain spin representations (see Appendix A) which SO(10) does not have – this is as crucial for Spin(10) Grand Unification as 1 the spinor representations of SU(2) = Spin(3) are to spin- 2 particles in quantum mechanics. CHAPTER 3. THEORY 20

In the Spin(10) GUT, first proposed by Georgi in 1974 [2], there is a natural choice for the 16 dimensional representation containing the right handed neutrino, namely the irreducible Weyl spinor representations. These are given by the direct sum of even and odd powers in the exterior algebra (see Appendix A)

V5 = (∧05 ⊕ ∧25 ⊕ ∧45) ⊕ (∧15 ⊕ ∧35 ⊕ ∧55) (3.19)

having dimensions 32 = (1 + 10 + 5) + (5 + 10 + 1) = 16 + 16 (3.20) Now there is a close connection between the SU(5) and Spin(10) GUTs and their fermion representations. First of all, SU(5) can be embedded as a subgroup in Spin(10) via a group homomorphism ψ, just like the Standard Model gauge group can be embedded in SU(5) by a homomorphism φ : GSM → SU(5). This gives the chain of subgroups

φ ψ GSM −→ SU(5) −→ Spin(10) (3.21) The irreducible representations of one fermion generation in the Standard Model can be written as a 16 dimensional direct sum F . Adding their antiparticles in F¯, GSM will act as unitary transformations on F ⊕ F¯, inducing a group homomorphism GSM → U(F ⊕ F¯) [2]. Similarly, both SU(5) and Spin(10) act as unitary transformations on V 5. It can be proved that the diagram

φ ψ GSM > SU(5) > Spin(10)

∨ ∨ ∨ U(φ) 1 U(F ⊕ F¯) > U(∧5) > U(∧5) commutes [2]. Let us now take a detour and look at a different route to go from GSM to Spin(10). The path will go over a unified theory based on the semisimple gauge group SU(2)×SU(2)×SU(4), which was first proposed by Pati and Salam. One of the main ideas in the Pati-Salam model is to extend the QCD color symmetry by a fourth dimension to SU(4) – here the colorlessness of leptons is considered as a fourth color. The other main idea is to create a symmetry between left and right handed fermions by adding another SU(2) factor, thus giving the gauge group GPS = SU(2)L×SU(2)R×SU(4)lc where lc is short for “lepton color.” Both these additional symmetries have to be broken down to those of the Standard Model. The group GPS will act on the vector space ¯ 32PS = (2L ⊗ 1R ⊕ 1L ⊗ 2R) ⊗ (4lc ⊕ 4lc¯ ) (3.22) which can be shown to be isomorphic to the one generational Standard Model representation: l F ⊕ F¯ −→ 32PS. In fact, there is a commutative diagram

GSM > GPS

∨ ∨ U(l) U(F ⊕ F¯) > U(32PS) From Lie theory we have the following Lie group isomorphisms:

Spin(4) =∼ SU(2) × SU(2) and Spin(6) =∼ SU(4) (3.23) CHAPTER 3. THEORY 21 where Spin(4) and Spin(6) are the double covers of SO(4) and SO(6), respectively. The obvious inclusion SO(4) × SO(6) → SO(10) (3.24) can be lifted to an inclusion

Spin(4) × Spin(6) → Spin(10) (3.25)

Spin(4) and Spin(6) have Dirac spinor representations V 2 and V 3, respectively, and the map ∧ : V 2 ⊗ V 3 → V 5 is an isomorphism. The following diagram can be shown to commute [2]:

Spin(4) × Spin(6) > Spin(10)

∨ ∨ U(∧) U(∧2 ⊗ ∧3) > U(∧5) ∼ There are also isomorphisms GPS = Spin(4)×Spin(6) and k : 32PS → ∧2 ⊗ ∧3 making the diagram

GPS > Spin(4) × Spin(6)

∨ ∨ U(k) U(32PS) > U(∧2 ⊗ ∧3) commute. Putting the last three diagrams together we get the commuting diagram

GSM > Spin(4) × Spin(6) > Spin(10)

∨ ∨ ∨ U(F ⊕ F¯) > U(∧2 ⊗ ∧3) > U(∧5) showing the Pati-Salam route from the Standard Model to the Spin(10) GUT. This diagram and the first SU(5) diagram can be united into a commuting cube:

GSM > SU(5)

> > GPS > Spin(10)

∨ ∨ U(F ⊕ F¯) > U(∧5)

> ∨ > ∨ U(∧2 ⊗ ∧3) > U(∧5)

It seems that the properties of the Standard Model are those which are given by SU(5) and the Pati-Salam model put together, both unified in the Spin(10) GUT. CHAPTER 3. THEORY 22

3.2 Neutrino Oscillation and Discrete Flavor Symmetries

A more thorough discussion of the pros and cons with Grand Unified Theories is saved for the next chapter, but at this point there is one disadvantage with GUTs we want to put the finger on. Contrary to the name, these attempts at unification might not be so “grand” after all, because they only follow the structure of the Standard Model and treat the fermions one generation at a time. Thus GUTs have no ambition of explaining the family structure of quarks and leptons. So even though GUTs do contain some interesting ideas, the question is whether the GUT project is doomed to fail from the beginning because of lack of ambition. A “Truly Unified Theory” [6] (or TUT for short) should be able to explain all the different elementary particles and their connections in a single mathematical structure with no ad hoc repetition of particle families. In this section we will look at an attempt to put the GUT structure in a slightly better light by adding another dimension of unification with the aid of a discrete group.3 While the GUT symmetry tries to unify the fermions within one family, this new family symmetry, given by a group GFAM, goes across the fermion generations. The total symmetry will be given by the direct product GGUT × GFAM. The structure of cross generational symmetry has its origin in neutrino physics. In the Standard Model neutrinos are massless, but several experiments the last couple of decades have shown, indirectly via neutrino oscillation, that at least one type of neutrino is in fact massive. The neutrinos in the Standard Model are linear combinations of neutrino mass states ν1, ν2, and ν3 with masses m1, m2, and m3, according to       νe Ue1 Ue2 Ue3 ν1  νµ  =  Uµ1 Uµ2 Uµ3   ν2  (3.26) ντ Uτ1 Uτ2 Uτ3 ν3

The unitary matrix U with coefficients is called the “lepton mixing matrix.” Unlike quarks, neutrinos mix strongly between generations. Neutrino oscillation refers to the fact that neutrinos can change lepton number (or “fla- vor”) when travelling over far distances, e.g. an electron neutrino can convert to a muon neutrino. Neutrino oscillation experiments only give information about the mass square dif- 2 2 2 2 ference ∆mij = mi −mj , so the actual masses are still unknown even though some ∆mij 6= 0, implying that not all three mi can be zero. There are experimental evidence from three types of neutrino oscillation. The first occurs in “atmospheric neutrinos,” i.e. neutrinos produced when cosmic rays are hitting the upper atmosphere. A shortage of atmospheric muon neutrinos was first confirmed by the Super- Kamiokande in 1998. The second type is “solar neutrinos,” where electron neutrinos from the Sun are converted into muon and tau neutrinos; this was confirmed by the Sudbury Neutrino Observatory in 2002. The third situation where neutrino oscillation has been observed is in neutrinos from nuclear reactors, e.g. at KamLAND in Japan. Now there are different suggested mixing patterns. In one of them, called trimaximal mixing, there is about the same√ probability of finding the mass state ν2 in any of the states νe, νµ and ντ ; here |Ue2| ≈ 1/ 3 and the mass state ν2 corresponds to a mass of at least m2 ≈ 0.008 eV. In another, called maximal mixing, ν3 contains νµ and ντ with approximately

3The exposition will follow [7]. CHAPTER 3. THEORY 23

√ equal probabilities. In that case, |Uµ3| ≈ |Uτ3| ≈ 1/ 2, giving a maximal mixing and oscillation of νµ and ντ . While the experimental results lead to some approximate mixings as mentioned above, there are some theoretical models with exact mixing patterns. One of these, known as tri- bimaximal (TB) mixing, make the assumption that

|Ue3| = 0, √ |Uµ3| = |Uτ3| = 1/ 2, √ (3.27) |Ue2| = |Uµ2| = |Uτ2| = 1/ 3.

The TB mixing has inspired several attempts of family unification with a discrete symmetry group. Even though some experiments (Daya Bay and RENO) have shown that |Ue3| 6= 0, thus ruling out the TB mixing model, we will still discuss the idea of discrete family symmetry. It might be a bad substitution for a True Unification, but since symmetries in different aspects have proven to be of fundamental importance, we still think such a discussion is worthwhile. The focus will be on non-abelian discrete groups with irreducible three dimensional rep- resentations, i.e. triplets. However, we will begin with an abelian group, namely the Klein four group, Z2 × Z2 (named after the mathematician Felix Klein, not to be confused with the physicist Oscar Klein). The Klein group enters when the neutrino mass matrix is rewritten ν,diag in a basis of diagonal charged leptons, denoted by mLL . We have ˜ T ν,diag ˜ ν,diag Kp,qmLL Kp,q = mLL (3.28) where  (−1)p 0 0  ˜ q Kp,q =  0 (−1) 0  (3.29) 0 0 (−1)p+q and p, q ∈ Z2. The corresponding relation for the non-diagonalized neutrino mass matrix ν mLL is T ν ν Kp,qmLLKp,q = mLL (3.30) where ∗ T Kp,q = U K˜p,qU (3.31) and U is the lepton mixing matrix. The (multiplicative) group of matrices K = {Kp,q : p, q ∈ Z2} is isomorphic to (the additive group) Z2 × Z2 via Kp,q 7→ (p, q). Denoting the generators ∼ of the factors in K = Z2 × Z2 by A1 and A2, respectively, in the case of TB mixing, these will have the appearance

 −1 2 2   1 0 0  1 A = 2 −1 2 ,A = − 0 0 1 (3.32) 1 3   2   2 2 −1 0 1 0

In this case it is a necessity that the charged leptons are (approximately) diagonal. The symmetry of the (squared) charged lepton mass matrix is given by the conjugate action of the matrix  1 0 0  − 2πi T =  0 e 3 0  (3.33)  2πi  0 0 e 3 CHAPTER 3. THEORY 24

The discrete family symmetry group is then generated by A1,A2 and T . However, when quarks are taken into account, the T symmetry is generally not exact. In what is called a direct model building approach, the family symmetry of TB mixing is determined by the semidirect product K o S3 (see Appendix A), which is isomorphic to S4. The total family symmetry group GFAM (e.g. S4) is broken down to the Klein symmetry of neutrinos by so called flavon fields aquiring a non-zero vacuum expectation value, in analogy with the Higgs mechanism. One indication that quarks and leptons are connected within a fermion generation is the fact that the smallest leptonic mixing angle is of the same magnitude as the largest quark mixing angle (the Cabbibo angle). This kind of symmetry is described by a GUT as discussed in the last section. A unification of all three generations would then be accomplished by combining the GUT and the family symmetry into the product group GGUT × GFAM. One example of such a unification is SU(5)×S4 describing TB mixing. To accomodate for a large experimental value of |Ue3|, an extra flavon field has to be added which is a singlet under S4.

3.3 A Note on SU(8) Grand Unification

To wrap this chapter up we would like to briefly mention yet another attempt at Grand Unification, this time with the gauge group SU(8). Such attempts are discussed in [12, 13, 14, 15]. In [12], several constraints for Grand Unification building are set up, and according to these requirements SU(8) is shown to be the most reasonable choice. One of these desired properties is that irreducible representations of the gauge group should not be repeated, but at the same time all fermions should be accomodated for. Thus, this model includes flavor unification within a simple gauge Lie group – no multiplication with a discrete group is needed – and so is an attempt of True Unification. All the left-handed leptons are in this SU(8) GUT put in the representation 8¯ ⊕ 28¯ ⊕ 56 (3.34) which, after breaking the SU(8) symmetry down to SU(5)×U(1)×SU(3), transforms as

8¯ = (5¯, 1) ⊕ (1, 3¯) (3.35) 28¯ = (10¯ , 1) ⊕ (5¯, 3¯) ⊕ (1, 3) (3.36) 56 = (10¯ , 1) ⊕ (10, 3) ⊕ (5, 3¯) ⊕ (1, 1) (3.37)

where the first entry in each pair gives the SU(5)×U(1) transformation, and the second entry is the color SU(3) part. This GUT model predicts the existence of eleven different quarks, six negatively charged leptons and ten neutral leptons. Another attempt at an SU(8) unification is considered in [13]. Here some quarks and leptons are fundamental, while others are assumed to be composed of so called hypercolored fermions – hypercolor (or technicolor [6]) is another SU(3) symmetry introduced in this model. The ambition of thisSU(8) “super GUT” is to include gravity as well as the electroweak and strong interactions, thereby making use of a suggested SO(8) supergravity mechanism. Chapter 4

Discussion

We started out the last chapter by mentioning some of the weaknesses and failures of the Standard Model. The rest of that chapter was devoted to some attempts of unification of particle physics beyond the Standard Model. What was not adressed was how well these attempts solve the problems of the Standard Model – that is the topic for this chapter. We will also see that when a certain theory might make a correct prediction of a certain phenomenon where the Standard Model runs short, e.g. quantization of electric charge, the alternative theory often comes equipped with some properties which have never been seen in experiments, e.g. proton decay. This unwanted extra luggage will also be dissected from time to another.

4.1 Grand Unified Theories

Let us begin our discussion with the prototypical Grand Unifed Theory of Georgi and Glashow, i.e. SU(5). If not otherwise mentioned, we here refer to [1]. A first advantage with SU(5) – and with all GUTs – compared to the Standard Model is the fact that a single gauge coupling constant is needed, thus reducing the number of unknown parameters from three to one. This is a consequence of the choice of gauge group as simple, rather than the semisimple Standard Model group U(1)×SU(2)×SU(3). If SU(5) would be a correct theory, we would also be one step closer to a desirable mathematical simplicity. (A problem that would still remain is why Nature would then favorize SU(5) among all simple Lie groups – at the end of the day there is nothing unique or special with SU(5) from a mathematical point of view. See section 5.3 for a more natural choice in this respect.) Unfortunately, as we will see in a short while, SU(5) Grand Unification does not quite reach the goal of experimental verification. On the contrary, there is a rather strong indication which rules out this whole theory – the prediction of proton decay with a much smaller lifetime than is seen in experiments. As was noted in chapter 3, SU(5) comes equipped with an additional twelve “leptoquark” bosons. These have charge ±4/3 (X/X¯) and ±1/3 (Y/Y¯ ), respectively, and have a mass equivalent to the energy of the GUT scale, i.e. at the order of 1016 GeV. In the SU(5) theory, leptoquarks make the transformation of a quark into a lepton possible, e.g. d¯ → e + X and u¯ → e + Y . It opens up the possibility that a proton decays via the exchange of a leptoquark to a π0 (quark content uu¯) and an electron. The theory predicts a lifetime for the proton of about 1030 years. However, the current lower experimental bound is 1033 years, thus making

25 CHAPTER 4. DISCUSSION 26 the SU(5) theory nothing more than a nice attempt at unification. Let us now turn to one of the great benefits of SU(5) – something which still makes the theory into a role model for other attempts at unification. The Standard Model gives no explanation why electric charge is quantized or why the magnitude of the proton and the electron charges are exactly the same. The SU(5) GUT explains these properties in a nice and natural way considering the representation structure of the theory. Recall that in (3.12) a down quark of three different colors were put together with a positron and an antineutrino in a five dimensional representation. When embedding the gauge group of the electroweak interaction, U(1)×SU(2), into SU(5), the photon, being one of the electroweak gauge bosons, will become an SU(5) gauge boson. This implies that the charge operator Q must be traceless [6]. Applied to the multiplet 5 in (3.12) this gives

Tr Q = 3Qd + Qe+ + Qν¯e = 0 (4.1)

Since the antineutrino is neutral, it follows that the charge of the d quark is exactly 1/3 of the electron charge. A similar argument can be used for the u quark, and is then repeated for the quarks of the second and third generation. Let us now turn to another good feature of SU(5) unification, namely the predicted value 2 of sin θW , where θW is the so called weak mixing angle (sometimes called the Weinberg angle after Stephen Weinberg, one of the creators of the electroweak unification). Recall from 1 2 3 chapter 3 that there are four fields in the electroweak theory: Bµ, Wµ , Wµ and Wµ . Two of 1 2 these, Wµ and Wµ , are combined into the electrically charged fields 1 1 W + = √ (W 1 − iW 2),W − = √ (W 1 + iW 2) (4.2) µ 2 µ µ µ 2 µ µ

± 3 corresponding to the charged vectors bosons W . The remaining two fields, Bµ and Wµ , are rotated into the fields Zµ and Aµ according to

 Z   cos θ − sin θ   W 3  µ = W W µ (4.3) Aµ sin θW cos θW Bµ where g g cos θ = 2 , sin θ = 1 (4.4) W p 2 2 W p 2 2 g1 + g2 g1 + g2 and g1, g2 are the gauge couplings of the electromagnetic and weak interaction, respectively [3]. 0 The fields Zµ and Aµ correspond to the neutral vector boson Z and the photon, respectively. By using the experimental values of the masses of the vectors bosons it can be shown that [3]

2 sin θW = 0.2315 ± 0.0004 (4.5)

2 The predicted value of sin θW in the SU(5) theory is 3/8 = 0.375, which at first sight does not look very impressive – it was actually rather disastrous for the theory until it was realized that this was the predicted value at the GUT scale, and has to be adjusted to the low energy scale of our present Universe [6]. Just as the coupling strengths depend on the energy scale 2 of renormalization µ (see equation (3.4)) as well as the GUT scale MSU(5), so does sin θW according to 3  109 M  sin2 θ (µ) = 1 − α log SU(5) (4.6) W 8 18π em µ CHAPTER 4. DISCUSSION 27 which gives 2 sin θW (µ) ≈ 0.21 (4.7) at the energy scale µ ∼ 1 GeV [6]. This rather good agreement with experimental results is one of the main reasons to believe in SU(5) unification [6]. Some other important quantities for a unifying theory to predict which also change with the energy scale are the masses of the fermions. In this case SU(5) does a poor job. At a GUT energy scale it gives md = me− , ms = mµ− and mb = mτ − . When breaking the symmetry and assuming the correct number of quark flavors (six as far as we know) the SU(5) theory gives the prediction mb = 5.3 GeV which is in good agreement with experiment. However, when turning to the mass of the strange quark, it is not so good, and the prediction for the mass of the d quark is far off. A different GUT could possibly do a better job by giving some additional contributions to the fermion masses [6]. Another possibility which would give very good predictions for the quark masses is to introduce another Higgs field of dimension 45, something which is done in Spin(10) unification [4, 6]. After these discussions of concepts depending on the energy scale, let us recall the first appearance of such ideas in this text, namely the running of the coupling constants in section 3.1. Taking a new look at Figure 3.1, it is obvious that the different lines do not intersect in a single point, indicating that SU(5) does not really unify the electroweak and strong interactions even at a high energy at the order of 1015 GeV. Now there are different ways to go to cure this illness. One of them is what is called supersymmetry, and originates from the same year as the SU(5) GUT, i.e. 1974 [1]. In a supersymmetric theory there is an additional symmetry between fermions and bosons – each particle will have a companion of the other particle type. For example, an electron has a spin one superpartner called a “selectron,” a photon is paired up with a spin half particle called a “photino” etc. However, the supersymmetry must be broken – if not, superpartners would have the same mass as the known particles, and no such superpartners have ever been detected [1]. So why is supersymmetry still today a very attractive idea to many working in the field of particle physics? One reason is that when taking the new particles into account, the running of the coupling constants can be made to meet at a single point (see Figure 4.1), thus making supersymmetric SU(5) a more attractive candidate for unification. Here the coefficients of (3.5 - 3.7) will get the corrections

1  2  ∆b = −2 − N (4.8) 3 4π 3 f 1  4 2 1  ∆b = − − N − N (4.9) 2 4π 3 3 f 3 H 1  2 1  ∆b = − N − N (4.10) 1 4π 3 f 5 H where NH = 2 is the number of Higgs fields [4]. Another reason for people not believing in GUTs to take supersymmetry into account is that the undetected superparticles could be considered as dark matter candidates. An alternative to supersymmetry is to extend the GUT group from SU(5) to Spin(10). In a Spin(10) GUT, symmetry breaking has to occur more than once into order to reduce Spin(10) to the Standard Model gauge group GSM = U(1)×SU(2)×SU(3). In this case the running of the three coupling constants does not have to meet at a single energy scale. While CHAPTER 4. DISCUSSION 28

Figure 4.1: A schematic picture of the running of the (inverted) coupling constants with supersymmetry – the couplings will now meet at a single point. Note the logarithmic scale on the energy axis (GeV). Here the solid line shows the electromagnetic coupling, the dashed line the weak coupling and the dotted line the strong coupling. this is an advantage, at the same time some predictive power of the model is lost since there are many different symmetry breaking patterns. We list a few of them here [4]:

Spin(10) → GSM (4.11)

Spin(10) → SU(5) → GSM (4.12)

Spin(10) → SU(4) × SU(2)L × SU(2)R → GSM (4.13) In the third case we recognize the left-right symmetric Pati-Salam model. The first breaking in each case occurs at the GUT scale, while the second breaking occurs at some lower in- termediate energy level. These two energy scales can be predicted from low energy coupling constants. (This, however, requires the use of the renormalization group equations which are beyond the scope of this treatment of unification in particle physics). The GUT scale in Spin(10) unification is higher than in the SU(5) case, high enough to make nucleon decay consistent with experimental values [4]. With the right choice of energy level for the interme- diate symmetry breaking, the value of the weak mixing angle can also be improved on. Yet another advantage of Spin(10) unification compared to SU(5) is that it comes equipped with massive neutrinos [4].

4.2 Discrete Flavor Symmetries

Let us also briefly discuss the attempts of unification with discrete family symmetry groups, in combination with GUTs. We will refer to our main source for neutrino physics, [7]. It is clear that recent experiments have ruled out many of the models based on discrete groups. It is also a fact that the models which have survived are rather complicated, lacking the kind of mathematical simplicity one could wish for in a theory with the ambition to be a substitution and improvement of the Standard Model. Still, the authors of [7] argue that the symmetry approach is still worth pursuing for a couple of reasons. One of these is that even though the theories are far from perfect, they will still be relatively good approximations of CHAPTER 4. DISCUSSION 29 a possibly correct theory, if such a theory exists. Another reason is that the use of discrete symmetries inspired by neutrino physics could still give some clue to the flavor problem, i.e. why the mixing parameters and the masses of the quarks and leptons are what they are. It might be a small chance, they further argue, that more clever theoretical work around the symmetries of neutrino physics could solve the whole flavor puzzle, and hopefully inspire young researchers to find something better than the rather complicated models proposed today. We will finish this chapter with a quotation from [7] which those young researchers might want to keep in mind when reading the outlook in the next chapter:

The history of physics, if it tells us anything at all, teaches us that symmetry and unification have always provided a guiding light in the path to understanding deep problems in physics. Chapter 5

Outlook

One could argue that GUTs – combined or not with discrete flavor symmetry – are even worse than the Standard Model itself. While GUTs reproduce the Standard Model at low energies, they also introduce new contradictions to experimental results such as the decay of protons and the existence of several bosons never observed. However, the idea of GUTs could still be worth pursuing if the ambition is set on the right level, i.e. to unify all elementary particles – bosons as well as fermions – and their interactions within a single mathematical framework. When Einstein created a new theory of gravity – General Relativity – he based his theory on taking the Equivalence Principle seriously, together with a new kind of symmetry – the invariance group of spacetime, given by special relativity – which had proved to be of fundamental importance [6]. This forced him to start from first principles and discard the idea of gravitational forces in favor of a curved spacetime. To reach a True Unification in particle physics – a theory which would have a fair chance to be unified with General Relativity – maybe we have to start from scratch in a similar fashion. Well, not totally from scratch. The principle of local gauge invariance should possibly be one of the corner stones of such a theory. But what else? A wonderful answer would be: choose the correct gauge group, i.e. the correct symmetry, and everything will follow if interpreted in the right way. To this day, no one seems to have fulfilled such a hope. But is it even possible? Maybe, maybe not. The price which has to be paid could be higher than most people are willing to accept. But still, in this chapter we will take a look at some ideas that might be fruitful to consider, or at least a shame to discard without a serious attempt. Some parts of this chapter will admittedly be of a more speculative nature than the rest of the topics covered so far. While GUTs also are speculations, these have been a part of the game for several decades and are by now very “established speculations,” and are considered by many in the field as quite interesting and reasonable. Some of the ideas introduced below are much less established, but hopefully just as interesting – and reasonable.

5.1 Substructure

An alternative direction to go which not many people take seriously these days is the idea of substructure, i.e. that leptons and quarks (and possibly also bosons) might not be elemen- tary particles, but may consist of one or several even more fundamental building blocks [1]. Probably the hierarchy of subparticles has to end somewhere, but the question is whether

30 CHAPTER 5. OUTLOOK 31 the bottom has been reached, or if there is yet another layer of reality beyond our present understanding of matter. In the 1960s when Gell-Mann and Zweig independently invented the quarks (Zweig calling them “aces”), it was known from experiments in the 1950s that protons where not pointlike particles, and so could possibly have substructure. However, for quite some time Gell-Mann himself was thinking of quarks as a convenient mathematical tool rather than actual physical objects [9]. As was touched upon at the end of chapter 3, the idea of further substructure was not considered too bizarre to people working on GUTs at the end of the 1970s [13]. Maybe substructure is necessary in order to find a Truly Unified Theory. If Gell-Mann and Zweig could take the step of introducing particles with one third the charge of the electron – at that time quite an extraordinary thought – maybe time has come to go full circle and complete the symmetry. This could possibly be accomplished by substituting the six leptons with six new light quarks. Instead of introducing new superheavy fermions which would naturally escape all experimental evidence, these new superlight quarks would also be very hard to detect, even indirectly. Furthermore, they could not only be combined to make the known leptons1, but could also be a potential source for dark matter when put together in new ways, either among themselves or mixed with the “old” quarks. It could also be a way to shed some new light over neutrino oscillation. If neutrinos are composed of quarks, neutrino oscillation might be understood in a way analogous to the decay of a neutron into a proton – one of the constituent quarks change flavor.

5.2 Unification of Quantum Numbers

In the Standard Model there is a large number of quantum numbers – spin, (weak) isospin, electric charge, hypercharge, three different color charges, lepton numbers, quark flavors, baryon number, parity etc. Just as we wish to unify all particles and describe them with a small number of elementary particles, it would be desirable to find a simple explanation for this long list of quantum numbers in terms of a few “natural” quantum numbers. (All of the listed numbers are not independent – e.g. electric charge, isospin and hypercharge are related to each other through the Gell-Mann-Nishijima formula – see chapter 2). While spin is intimately connected with the symmetry of spacetime (via spin representations of (the double cover of) the Lorentz group), most of the other numbers are considered “internal”, not to be mixed up with spacetime (here parity is an exception). However, some of the properties they describe are very present in the physical four dimensional spacetime reality. Take, for example, electric charge which has direct consequences for everyday life. Should it not be possible to give electric charge a simple explanation in terms of all there is – the geometry of spacetime? But electric charge is quantized and spacetime is not. Or wait a second, maybe it is – in terms of dimensions. So what if we introduced four “dimensional quantum numbers” t, x, y and z, with possible values of 0, 1 or −1? These should indicate the local orientation of spacetime, and each particle type should be given by a vector, e.g. u = (0, 0, 1, 1) could be an up quark. In this scheme, electric charge could for instance be given by Qdim = x+y+z, where the unit is the smallest quark charge which seems to be the fundamental unit rather than the proton charge. For the hypothesized up quark representation, this would imply Qdim(u) = 2.

1Actually, it follows from Heisenberg’s uncertainty principle that in order for such small particles to have subparticles, there must exist a force much stronger than the strong force to hold these subparticles together. If this is possible to accomplish within the framework of GUTs is not very clear. CHAPTER 5. OUTLOOK 32

(Qdim is then related to ordinary charge by Qdim = 3Q.) An elementary particle is in this sense not a pointlike object – how could a zero dimensional object come equipped with any quantum numbers at all? Instead, an elementary particle is a local orientation of spacetime. The antiparticle of a particle has the opposite orientation, e.g.u ¯ = (0, 0, −1, −1). If electric charge could be given a geometrical interpretation, what about color charge? The “color” terminology is obviously very ad hoc and it would be desirable to find a more natural notion with a more direct interpretation in the physical reality. Now the idea of color and complement color, colorlessness etc. is nicely captured by the exterior algebra over a three dimensional space, together with Hodge duality [2]. In [1], the wave function of a baryon is written ψ = ψ(space)ψ(spin)ψ(flavor)ψ(color) where the color part has to be a singlet under SU(3) and so has to be of the form √ ψ(color) = (rgb − rbg + gbr − grb + brg − bgr)/ 6

The product should here be interpreted as a tensor product and so this can be written as 1 ψ(color) = √ r ∧ g ∧ b 6 If we now replace “colors” with spatial dimensions x, y, and z, and instead consider a basis dx, dy, dz for the dual space, the “color” part of the wave “function” is the differential three form 1 ψ(color) = √ dx ∧ dy ∧ dz 6 and we end up with a wave (volume) form instead of a wave function. From a mathematical point of view this is quite natural since we usually want to integrate it over some part of a space(time) manifold. In the language of “color” this means that a baryon must have three different colors of the quarks – to integrate a one or two form does not make any sense in this context. (For a meson with a quark-antiquark pair, the volume form is given by Hodge duality since dx ∧ ∗dx = dx ∧ dy ∧ dz.)

5.3 A Spin(8) Truly Unified Theory

In the discussion of SU(5), Spin(10) and the Pati-Salam model in chapter 3, the double cover of three rotational groups in even dimensions showed up: Spin(4), Spin(6) and Spin(10). One might wonder if the missing element in this sequence, i.e. Spin(8), has any significant role to play in a unifying theory. In 1986 there was an attempt to build a GUT with Spin(8) as gauge group [16]. The focus in that paper was to try to understand the size of the different masses of elementary particles by looking at the structure of the octonion algebra and its connection to Spin(8). As is noticed in the same paper, this attempt unfortunately leads to a dead end since at least one of the predicted values is way off the experimental value. In [6] the possibility of building a GUT on Spin(8) is ruled out from the start. The reason mentioned is that the only rank four group which has complex representations to accomodate for the fermions in the Standard Model is SU(5), and so it is argued, SU(5) is the only choice for a rank four gauge group. CHAPTER 5. OUTLOOK 33

But what if Spin(8) has some other properties which could compensate for the lack of complex representations? Among all simple Lie groups, Spin(8) is a very special – and unique – structure: it is the only group where the (half) spin representations have the same dimension as the fundamental vector representation: they are all eight dimensional and equivalent. This special symmetry, which is determined by the discrete group S3, is called triality. (This is not to be confused with trinification – an attempt at unification with the semisimple gauge group SU(3)×SU(3)×SU(3); see e.g. [17]). One idea to make use of triality is to put the eight gluons in the triality representations. But what should then fill out the twenty eight dimensional adjoint representaion? Well, first of all the four electroweak bosons corresponding to the four elements of the Cartan subalgebra. In the remaining twentyfour dimensions, given by twelve pairs of root spaces, there should be twelve quarks and their antiparticles – the six known quarks and six new quarks building up the leptons and possibly some unknown particles. Thus, bosons and fermions will have different roles to play in the mathematical structure of so(8) – the Lie algebra (adjoint representation) of Spin(8). One advantage of this model is that it will actually explain the particle content of the Standard Model, and put all the fermions in the same representation, so there is no need for a threefold repetition of the different generations. Also, for any other GUT there is no explanation why that specific gauge group should be choosen. For Spin(8) such a reason does exist – it is, in a certain sense, the most symmetric choice. And Nature seems to like symmetry. A disadvantage is that in gauge theories, and more specifically Yang-Mills theories, the adjoint representation is supposed to contain massless bosons only. The mass problem could probably be solved by a twentyeight dimensional Higgs field, breaking the symmetry, simi- larly to the SU(5) case. But what about the fermions? Admittedly, this means that some basic notions have to be given another deep thought – is there a different gauge theoretical generalization of electromagnetism other than Yang-Mills theory? At the same time it would be quite attractive of a theory if the choice of gauge group could also give the fermions to put in the Lagrangian, instead of first guessing (from observation or otherwise) the fermion content, then require local gauge invariance just to get some additional bosons to compensate for the lack of symmetry. Instead, symmetry would be the origin of all fields involved, and the Lagrangian would be manifestly gauge invariant. Chapter 6

Conclusion

During the twentieth century, particle physics developed into a cornerstone of modern physics, culminating in the Standard Model at the beginning of the 1970s. Even though this theory has proved to be of excellent agreement with experimental results, it is still incomplete in several respects. For example, it does not explain the existence of massive neutrinos or dark matter. The Standard Model also has some structural problems, one of them being the fact that there are too many free parameters which have to be deduced from experiments rather than being explained by theory, e.g. the different masses of the elementary particles, and the gauge and Yukawa couplings. From a mathematical point of view the Standard Model also have some struggles – its obvious lack of simplicity is definitely a challenge for the next generation of theorists to improve on. It has been our aim in this bachelor thesis to discuss some possible theories of unification in particle physics beyond the Standard Model. But before these ideas were developed, the foundation of all particle theories was introduced – the principle of local gauge invariance – which has proved to be of fundamental importance in modern physics, especially in the context of elementary particles and their interaction. Furthermore, an introduction to the Standard Model itself was given. Two main ideas beyond the Standard Model was introduced, the first of them being Grand Unified Theories, where the three interactions (electromagnetic, weak and strong) is substituted by a single symmetry, which is supposed to have governed the physics in the very early Universe, and later was broken into the different gauge groups of the Standard Model. While some of these theories solve a few of the problems with the Standard Model, such as the quantization of charge, they introduce other problems not in agreement with experiments, e.g. the decay of protons. Another failure of Grand Unified Theories is that they do not even try to give an explanation of the fermion generational pattern in the Standard Model – any serious attempt of a Truly Unifed Theory of elementary particles should have a solution to this problem. This leads us to the other main idea of unification introduced in the text, namely dis- crete family symmetry. This idea originates from neutrino physics, and while Grand Unified Theories have not seen much of a development the last years, neutrino physics is a highly active research field with new generations of experimental devices being set up around the world. Where Grand Unified Theories only aim at unification within a fermion family, the theory of family symmetry tries to unify the different families with the use of a discrete group. However, recent experiments have ruled out many of the theoretical models in this area, and

34 CHAPTER 6. CONCLUSION 35 so some fresh perspectives are warmly welcome from a new generation of theorists who may be inspired by the latest experimental work. Finally we also gave some possible paths to go in the future in the quest for a unification in particle physics. These ideas were of a more speculative nature but are, in our opinion, nevertheless worth pursuing. Since the creation of the Standard Model there have been several major breakthroughs on the experimental side, e.g. the discovery of the tau particle, the top quark, and the Higgs boson, as well as neutrino oscillation, implying massive neutrinos. On the theoretical side it is more difficult to judge about the development since it often takes a long time before new theories can be experimentally verified, one example being the discovery of the Higgs boson. It might be the case that some of the theories suggested the last few decades will turn out to be true in fifty years from now. At the same time there is a risk that none of these will do the work. In our opinion it is crucial that alternative ideas of a less mainstream character are taken seriously and are allowed to be thoroughly investigated. After all, the price which has to be paid in terms of time, energy and other resources should be rather small compared to all the effort put in some of the theories of the last forty years which still lack empirical motivation. Appendix A

Groups, Algebras, and Representations

In this appendix we will try to give a brief introduction to the mathematical notions in the main text. Instead of trying to give a thorough treatment of each topic, which will fail anyway, we will give some basic definitions and examples, and include some references for the interested reader to dive into. The style of this (and the next) appendix will be slightly more formal and will make use of more mathematical symbols than the rest of the thesis.

A.1 Groups

Disputably the most profound concept of modern physics is symmetry, which mathematically is described by the algebraic structure of a group. Even though many physicists during the first half of the twentieth century felt uncomfortable with the growing impact of group theory on physics, referring to it as “Gruppenpest” (German for “group plague”), its deep relation to physical theories has grown ever since, and is today of such central importance that physics would be very difficult – not to say impossible – to imagine without it. Starting with space- time symmetry in special relativity and rotational symmetry of spin half states in quantum mechanics, group theory became of central importance to the Standard Model of particle physics. The Standard Model is built upon the principle of local gauge symmetry, where one abelian and two non-abelian Lie groups are combined into the semisimple gauge group U(1)×SU(2)×SU(3). The further attempts of unification discussed in this thesis – Grand Unified Theories and discrete family symmetry – are directly built on the group concept. Therefore we will in this section give an overview of the groups involved.

Let us start with the general definition.

Definition. A group is a pair (G, ◦) where G is a set and ◦ is a binary operation on G with the following properties: 1. a ◦ (b ◦ c) = (a ◦ b) ◦ c for all a, b, c ∈ G. 2. There exists an element e ∈ G such that e ◦ a = a ◦ e = a for all a ∈ G. 3. For every a ∈ G there is an element b ∈ G such that a ◦ b = b ◦ a = e. The first property says that the group operation is associative, while the element e in (2) is called the identity element of the group (it can be shown to be unique). The element b in

36 APPENDIX A. GROUPS, ALGEBRAS, AND REPRESENTATIONS 37

(3) is called the inverse of a and is in multiplicative notation denoted by a−1 (in additive notation −a is preferable). If in addition a◦b = b◦a for all a, b ∈ G, the group G is said to be abelian (after the Norwegian mathematician Niels Henrik Abel). When the group operation is understood from context, it is often dropped from the notation and the underlying set G is denoting the group.

Example. (Z, +), the integers with addition as group operation, is an abelian group of in- finite order, i.e. with an infinite number of elements. The identity element is 0, and the inverse of an integer n is −n.

More examples of interest will show up in the sections to come.

A.1.1 Finite and Discrete Groups

The group (Z, +) is an infinite discrete group, i.e. a group with the discrete topology. This is in contrast to Lie groups (see section A.1.4) which are sometimes referred to as continuous groups. (To be precise, finite and countably infinite groups are actually zero dimensional Lie groups with the discrete topology.) The discrete groups found in physics are often finite, i.e. having a finite number of elements.

Example. (Z2, ⊕), the integers modulo 2, where the set Z2 consists of the congruence classes [0] and [1], and the group operation is defined by [n]⊕[m] = [n+m]. This is a finite abelian group.

Example. (S3, ◦) is the group of permutations of the set {1, 2, 3}, i.e. the elements of S3 are all bijections from {1, 2, 3} to itself, and the group operation is composition of functions. The elements of a permutation group are often written in cyclic notation, e.g. (1 2) is short for 1 7→ 2, 2 7→ 1 and 3 7→ 3. The order of S3 is 6:

S3 = {(1), (1 2), (1 3), (2 3), (1 2 3), (1 3 2)} (A.1)

S3 is an example of a non-abelian group. For example, (1 3)(1 2) = (1 2 3) but (1 2)(1 3) = (1 3 2). S3 is isomorphic to – have the same group structure as – the symmetry group of an equilateral triangle (the dihedral group D3), where (1 2), (1 3), (2 3) correspond to reflections, while (1 2 3), (1 3 2) correspond to rotations by 2π/3.

A.1.2 Group Actions Given a group G and a set S (which might be equipped with some structure such as a vector space or a group), the group G can act on S. This means that there is a group homomorphism (i.e. a map which preserves the group strucure but does not have to be one-one or onto) from G to the group of bijections on S. Points of S which can be connected by the action of the group are said to be in the same orbit.

2 Example. Let G be the group of all rotations of the plane R around the origin. (This is an example of a Lie group denoted by SO(2), or, working in the complex number plane C, U(1) – see section A.1.4.) The orbits of this action are the circles centered at the origin.

Example. When a group K acts on a group H, the homomorphism should take values in APPENDIX A. GROUPS, ALGEBRAS, AND REPRESENTATIONS 38

Aut(H), the group of automorphisms of H – its elements are bijective homomorphisms (i.e. isomorphisms) from the group H to itself. If H and K are both subgroups of some group G, it makes sense to “multiply” elements of H and K, and an example of a group action is given when K acts on H by conjugation, i.e. for k ∈ K and h ∈ H the action is k.h = khk−1.

A.1.3 Direct and Semidirect Products of Groups Given two groups (G, ◦) and (H, ) their direct product (G × H,?) is defined by using the group operations of G and H on each factor:

(g1, h1) ? (g2, h2) := (g1 ◦ g2, h1  h2) (A.2)

Example. Let G = H = Z2. Then the direct product is the Klein four group Z2 × Z2 (after the German mathematician Felix Klein).

If H and K are groups and there is an action of K on H (denoted by k.h), the semidi- rect product of H and K, denoted by H o K, is the set H × K equipped with the operation (using juxtaposition for group operations)

(h1, k1)(h2, k2) = (h1(k1.h2), k1k2) (A.3)

Example. Zn o Z2, where Zn is the cyclic group of order n (isomorphic to (Zn, ⊕) but written with multiplicative notation), is isomorphic to the dihedral group Dn (the symmetry group −1 of a regular n-gon). Here the nonidentity element c of Z2 acts on Zn by inversion: c.b = b . In terms of generators and relations this group can be written as

ha, b | a2 = e, bn = e, aba−1 = b−1i (A.4)

A.1.4 Lie Groups If groups in general are important to physics, the groups which play the most central role are Lie groups (named after the Norwegian mathematician Marius Sophus Lie). For example, the gauge group of the Standard Model is a direct product of the three Lie (matrix) groups U(1), SU(2) and SU(3). Lie groups describe continuous symmetries, e.g. U(1) is the symmetry group of a circle, in contrast to finite groups which describe discrete symmetries.

Definition. A Lie group is a group which is also a differentiable manifold such that the group operation and the inverse map (a 7→ a−1) are smooth functions.

(For a very brief introduction to differentiable manifolds, see Appendix B.) Most of the Lie groups of interest in physics are matrix groups.

iθ 1 Example. U(1) = {e ∈ C | θ ∈ [0, 2π)}, i.e. the unit circle S in the complex plane. This group is isomorphic to the special orthogonal group SO(2) in two dimensions. Note that both have real dimension 1 – Lie groups are by definition real manifolds.

† Example. SU(n) = {U ∈ Mn(C) | U U = I, det U = 1}, i.e. all unitary n × n matrices of determinant 1. In the case n = 2, we get the group SU(2) which is isomorpic to the group APPENDIX A. GROUPS, ALGEBRAS, AND REPRESENTATIONS 39 of unit quaternions, which is also given by the three dimensional unit sphere S3 in four di- 4 mensions (H = R ), so SU(2) has dimension 3.

n Example. Spin(n) is the double cover of the group SO(n) of rotations in R . This means (skip- ping the topological details) that there is a two-to-one correspondence between the points of Spin(n) and SO(n). For example, SU(2) is isomorpic to Spin(3), the double cover of SO(3). A spin group can be realized as a certain group embedded in a Clifford algebra, and has spinorial representations (see sections A.2.2 and A.3) which do not descend to representations of the special orthogonal group covered by the spin group.

A.2 Algebras

There are several kinds of algebras which show up in physics. In this section we will look at a few of the most important – Lie algebras, Clifford algebras, tensor algebras and exterior algebras. Lie algebras give the local structure of Lie groups and are easier to study than the groups, but still give a lot of information of the groups, e.g. their representations. The Dirac algebra of gamma matrices, which is crucial to the formulation of the Dirac equation, is an example of a Clifford algebra. As mentioned in the last section, Clifford algebras are also important for the understanding of spinorial representations, which in turn require the notion of exterior algebras. Exterior algebras are subalgebras of tensor algebras and also play an important role in the theory of differentiable manifolds.

Definition. An algebra A is a vector space equipped with a bilinear map A × A → A, (a, b) 7→ ab, called multiplication. If the multiplication is associative (commutative) the algebra is said to be associative (commutative).

Example. The set Mn(C) of square matrices is an associative, but non-commutative, algebra.

A.2.1 Lie Algebras Before giving the connection to Lie groups, let us start with the definition and a few examples.

Definition. A Lie algebra g is an algebra where the multiplication (a, b) 7→ [a, b] (called the Lie bracket) is (1) anti-symmetric and (2) satisfies the Jacobi identity:

1.[ b, a] = −[a, b] for all a, b ∈ g.

2.[ a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0 for all a, b, c ∈ g.

3 Example. g = R , where [~u,~v] = ~u × ~v, is a Lie algebra.

Example. Given an associative algebra A, we will get a Lie algebra gA by defining the Lie bracket in gA as the commutator of elements in A:[a, b] = ab − ba.

Example. gln(C), the general linear algebra over C, is the Lie algebra given by the asso- ciative algebra Mn(C) according to the last example. APPENDIX A. GROUPS, ALGEBRAS, AND REPRESENTATIONS 40

The set of all (smooth) left-invariant vector fields on a Lie group G is a Lie algebra, de- noted by Lie(G) or g, and is called the Lie algebra of G. It can be proved that Lie(G) is isomorpic (as Lie algebras) to TeG, i.e. the tangent space to G at the identity element.

n Example. The Lie group GLn(R) consists of all invertible linear maps from R to itself. The Lie algebra Lie(GLn(R)) is isomorphic to gln(R).

If two Lie groups G and H are isomorpic, so are their Lie algebras g and h. Also, if G˜ covers G, then Lie(G˜) is isomorphic to Lie(G).

Example. Since Spin(n) is a covering space of SO(n), their Lie algebras spin(n) and so(n) are isomorphic.

A.2.2 Clifford Algebras A Clifford algebra (named after the English mathematician William Kingdon Clifford) is an associative algebra with a unit element 1 which is determined by a vector space V equipped with a quadratic form Q. It is denoted by Cl(Q, V ) and is the free algebra generated by V with the condition v2 = Q(v)1 (A.5) Here the product to the left is the algebra multiplication. In most cases of interest to physics, V is a vector space over R or C. The identity (A.5) can then be replaced by uv + vu = 2hu, vi1 (A.6) where 1 hu, vi = (Q(u + v) − Q(u) − Q(v)) (A.7) 2 is the symmetric bilinear form associated with Q. Over the complex numbers all quadratic n forms are equivalent and so there is a unique Clifford algebra Cln(C) determined by V = C . When n = 0 we get Cl0(C) = C.

n A quadratic form on V = R is equivalent to some 2 2 2 2 Q(v) = v1 + ... + vp − vp+1 − ... − vp+q (A.8) where n = p + q and v = (v1, . . . , vn). V then has the signature (p, q). Minkowski space 4 1,3 in special relativity is R with signature (1,3), sometimes written R . The Clifford algebra p,q associated to R is denoted Clp,q(R). ∼ ∼ ∼ Example. We have the isomorphisms Cl0,0(R) = R, Cl0,1(R) = C and Cl0,2(R) = H.

If {e1, . . . , en} is an orthogonal basis for (V,Q), i.e.

hei, eji = 0 for i 6= j, and hei, eii = Q(ei) (A.9) a basis for Cl(V,Q) is

{ei1 ei2 . . . eik | 1 ≤ i1 < i2 < . . . < ik ≤ n and 0 ≤ k ≤ n} (A.10) APPENDIX A. GROUPS, ALGEBRAS, AND REPRESENTATIONS 41 and n X  n  dim Cl(V,Q) = = 2n (A.11) k k=0 Example. The most famous Clifford algebra in physics is generated by the Dirac gamma matrices  1 0 0 0   0 0 0 1   0 1 0 0   0 0 1 0  γ0 =   , γ1 =   ,  0 0 −1 0   0 −1 0 0  0 0 0 −1 −1 0 0 0 (A.12)  0 0 0 −i   0 0 1 0   0 0 i 0   0 0 0 −1  γ2 =   , γ3 =    0 i 0 0   −1 0 0 0  −i 0 0 0 0 1 0 0 which satisfy the anti-commutation relations

{γi, γj} = γiγj + γjγi = 2ηij (A.13)

where η is the matrix of a quadratic form of signature (1, 3).

A.2.3 Tensor Algebras and Exterior Algebras There are many situations in physics where tensors and tensor products play an important role. One example is in quantum mechanics where the Hilbert space of a composite system is given by the tensor product of the Hilbert spaces of the parts, e.g. when two spin half 2 2 particles are put together the Hilbert space will be C ⊗ C .

The general definition of the tensor product V ⊗ W of two vector spaces V and W is rather abstract, making use of a universal property which informally says that bilinear maps from V × W are “linearized” when turned into maps from V ⊗ W . When V and W are finite dimensional, their tensor product is more easily understood in terms of bases. If a ba- sis for V is {e1, . . . , en} and a basis for W is {f1, . . . , fm}, a basis for V ⊗ W is given by {ei ⊗ fj | i = 1, . . . , n, j = 1, . . . , m}. Thus

dim V ⊗ W = dim V · dim W (A.14)

2 Example. For a spin half particle, the basis vectors of C are sometimes denoted by | ↑i and | ↓i. The composite system of two spin half particles will have the basis vectors | ↑i ⊗ | ↑i, 2 2 | ↑i ⊗ | ↓i, | ↓i ⊗ | ↑i and | ↓i ⊗ | ↓i and dim C ⊗ C = 2 · 2 = 4.

Let V ⊗k = V ⊗ V ⊗ ... ⊗ V , with k factors V .A tensor of type (k, l) over V is an element of the tensor product V ⊗k ⊗ (V ∗)⊗l. What physicists often refer to as a tensor, mathematicans call a tensor field, i.e. a tensor valued function defined on a (smooth) manifold (see Appendix B). For finite dimensional vector spaces, an equivalent way of looking at tensors is as mul- tilinear functions on V ⊗k ⊗ (V ∗)⊗l. With this approach, those tensors which take the value 0 when at least two arguments are equal are called alternating tensors, and the subspace of V ⊗k consisting of alternating tensors is denoted by Vk(V ). APPENDIX A. GROUPS, ALGEBRAS, AND REPRESENTATIONS 42

1 V2 Example. If v, w ∈ V , then v ∧ w = 2 (v ⊗ w − w ⊗ v) is an element of (V ). Similarly, if u, v, w ∈ V , then 1 u ∧ v ∧ w = (u ⊗ v ⊗ w − u ⊗ w ⊗ v + v ⊗ w ⊗ u − v ⊗ w ⊗ u + w ⊗ v ⊗ u − w ⊗ u ⊗ v) (A.15) 3! is an element of V3(V ).

The tensor algebra T (V ) over V is defined by

∞ M T (V ) = V ⊗k (A.16) k=0 where the multiplication of “simple” elements is given by tensor product and is then extended ⊗0 linearly. (V is defined to be the ground field, e.g. R or C.) Likewise, the exterior algebra V(V ) over V is defined by ∞ ^ M (V ) = Vk(V ) (A.17) k=0 While the tensor algebra T (V ) is infinite dimensional, dim V(V ) = 2n where n = dim V : for k > n the exterior product Vk(V ) is the zero vector space, and for 1 ≤ k ≤ n, a basis for Vk(V ) is

{ei1 ∧ ei2 ∧ ... ∧ eik | 1 ≤ i1 < i2 < . . . < ik ≤ n and 0 ≤ k ≤ n} (A.18)  n  so dim Vk(V ) = . As vector spaces, V(V ) and Cl(V,Q) are isomporphic via e ∧ e ∧ k i1 i2

... ∧ eik 7→ ei1 ei2 . . . eik .

4 Example. Let V = C . Then

^ V0 V1 V2 V3 V4 4 6 4 (V ) = (V ) ⊕ (V ) ⊕ (V ) ⊕ (V ) ⊕ (V ) = C ⊕ C ⊕ C ⊕ C ⊕ C (A.19)

having dimension 24 = 16.

A.3 Representations and Spinors

A.3.1 Linear Representations In many cases of interest to physics, groups and algebras act as linear transformations on a vector space – sometimes this space carries some further structure, e.g. an inner product with some properties making the space into a Hilbert space (this is the case in quantum mechanics).

Definition. Let G be a group. With a (linear) representation of G we mean a pair (V, φ) where V is a vector space and φ : G →GL(V ) is a homomorphism from G to the group GL(V ) of automorphisms of V .

Often one of V or φ is understood from context, so V or φ alone are frequently referred n to as a representation of G. When a basis for V is given, e.g. the standard basis of V = R APPENDIX A. GROUPS, ALGEBRAS, AND REPRESENTATIONS 43

n or V = C , GL(V ) can be identified with the group of invertible n × n matrices (with real or complex entries) – for g ∈ G, φ(G) is then a matrix. This is usually the way to think about representations – the action of G on V is given by matrix multiplication.

3 3 Example. Let S3 act on V = R or V = C in the following way:  1 0 0   0 1 0   0 0 1  (1) 7→  0 1 0  , (1 2) 7→  1 0 0  , (1 3) 7→  0 1 0  , 0 0 1 0 0 1 1 0 0 (A.20)  1 0 0   0 0 1   0 1 0  (2 3) 7→  0 0 1  , (1 2 3) 7→  1 0 0  , (1 3 2) 7→  0 0 1  0 1 0 0 1 0 1 0 0 under φ. For instance, when (1 2) acts on the basis vector e1 we get

 0 1 0   1   0  (1 2).e1 = φ((1 2))e1 =  1 0 0   0  =  1  = e2 (A.21) 0 0 1 0 0

Thus, S3 acts on V by permuting the basis vectors.

As mentioned in an earlier section, the groups of most interest in physics are Lie groups, and most Lie groups of interest are matrix groups. A representation of the group is then given by the natural action defined by matrix multiplication. This representation is called the defining or fundamental representation.

Example. The matrix Lie group SO(2) is given by all matrices which by a multiplication 2 give a rotation around the origin in two dimensional space R , i.e.  cos θ − sin θ   SO(2) = | θ ∈ [0, 2π) (A.22) sin θ cos θ

2 Its fundamental representation acts by matrix multiplication on vectors in R , e.g.  cos 45◦ − sin 45◦   1  1  1  = √ (A.23) sin 45◦ cos 45◦ 0 2 1

◦ rotates the basis vector e1 by 45 counterclockwise.

† Given a square matrix A ∈ Mn(C), you get the hermitian conjugate of A, denoted by A , by T taking the complex conjugate of each element of A . A matrix U ∈ Mn(C) having the prop- erty U †U = UU † = I is called unitary. Let U(V ) be the group of unitary transformations of V . If G is a group, a homomorphism φ : G → U(V ) is called a unitary represention of G on V .

Another important notion in the area of representation theory is that of irreducibility.

Definition. A representation (V, φ) is called irreducible if there is no non-trivial subspace of V which is closed under the action of all φ(g), g ∈ G. APPENDIX A. GROUPS, ALGEBRAS, AND REPRESENTATIONS 44

If such a subspace does exist, the representation is reducible. If the complementary subspace also is closed under the action of all φ(g), g ∈ G, the representation is completely reducible – the matrices φ(g) can then be written in block diagonal form. When trying to determine all representations of a given group, it is sufficient to find all irreducible representations – these are the fundamental building blocks in terms of which all representations of the group can be given.

Many of the notions given above for group representations can be carried over to repre- sentations of algebras, e.g. Lie algebras. The group GL(V ) of automorphisms then has to be replaced by the Lie algebra gl(V ).

Definition. A representation of a Lie algebra g is a pair (V, ϕ) where V is a vector space and ϕ : g → gl(V ) is a homomorphism of Lie algebras, i.e. ϕ preserves the Lie bracket:

ϕ([a, b]) = [ϕ(a), ϕ(b)] for all a, b ∈ g (A.24)

The vector space V on which g acts is sometimes called a g-module. As in the case of Lie groups, the fundamental representation of a Lie algebra acts on V by matrix multiplication. The most important representation of a Lie algebra is the adjoint representation, where the Lie algebra acts on itself (i.e. its underlying vector space) by commutation.

Definition. The adjoint representation of the Lie algebra g is the homomorphism ad: g → gl(g) given by ad(a)(b) = [a, b].

The Lie algebra homomorphism ad(a) ∈ gl(g) is often denoted by ada. There is also an adjoint representation of a Lie group G on its Lie algebra g = Lie(G) given by conjugation: the Lie group homomorphism Ad: G →GL(g) is defined by Ad(A)(X) = AXA−1, and Ad(A) is often written AdA. The task of finding all finite dimensional irreducible complex represen- tations of a Lie group can be carried over to determining all finite dimensional irreducible complex representations of its Lie algebra.

A.3.2 Spinors A crucial feature of quantum mechanics is that leptons and quarks are spin 1/2 particles, and 2 their spin state is represented by a vector in C (more accurately, it is a point in the projective 2 space P (C ), but a discussion of this is beyond the scope of this thesis). To rotate such a state it is not enough to use a representation of the group SO(3), but what is needed is a spin representation of its double cover Spin(3) = SU(2). Thus, such particles are called spinors. In this section we will try to understand how these spinor representations work – they also show up in different attempts of unification discussed in the main text, where Spin(4), Spin(6) and Spin(10) play a key role in a Grand Unified Theory, and the representation theory of Spin(8) has some striking symmetry which might be of some use in future attempts of unification.

The groups Spin(n) are intimately connected with the theory of Clifford algebras, so let us take a closer look at some features of these. In this section Cln will refer to the Clifford n algebra given by R with a positive definite quadratic form (this was denoted by Cln,0(R) APPENDIX A. GROUPS, ALGEBRAS, AND REPRESENTATIONS 45

in section A.2.2). First, there is the canonical automorphism α : Cln → Cln which on basis elements is defined by k α(ei1 ei2 . . . eik ) = (−1) ei1 ei2 . . . eik (A.25) and which introduces the following Z2-grading on Cln:

+ − Cln = {u ∈ Cln | α(u) = u}, Cln = {u ∈ Cln | α(u) = −u} (A.26)

Second, we have the conjugation Cln 3 a 7→ a¯ ∈ Cln which on basis elements is defined by

k ei1 ei2 . . . eik = (−1) eik eik−1 . . . ei1 (A.27)

× Conjugation is an anti-homomorphism in the sense that xy =y ¯x¯ for x, y ∈ Cln. Now let Cln be the group of invertible elements in Cln. The Clifford group Γn is the subgroup

× −1 n n Γn = {u ∈ Cln | α(u)xu ∈ R for all x ∈ R } (A.28)

The spinor group Spin(n) will be a certain subgroup of the Clifford group. But first we have to define the pinor group Pin(n)1, which is the kernel (i.e. all elements mapped to the identity) × of the map ν :Γn → R , ν(u) = uu¯. Finally we can set

+ Spin(n) = Pin(n) ∩ Cln (A.29)

For n ≥ 3, Spin(n) is simply-connected and so is the universal covering space of SO(n).

In order to find the irreducible representations of Spin(n) we first note that its Lie algebra ∼ + spin(n) = so(n) can be embedded in Cln in the following way:    X  so(n) = tijeiej | tij ∈ R (A.30) 1≤i

Since our main interest is in cases where n is even, we will make such an assumption from n 0 0 now on. We then have the decomposition R = W ⊕W where W and W are complementary ∼ V n/2-dimensional subspaces. It can be shown that Cln = End( (W )) where End(V ) is the associative algebra of all linear transformations of the vector space V . We also have the decomposition V(W ) = Veven(W ) ⊕ Vodd(W ) into a sum of even and odd exterior powers, which leads to an isomorphism

+ ∼ Veven Vodd Cln = End( (W )) ⊕ End( (W )) (A.31)

From this we get an embedding of Lie algebras

+ ∼ Veven Vodd so(n) ⊂ Cln = gl( (W )) ⊕ gl( (W )) (A.32) which gives the half-spin representations

S+ = Veven(W ),S− = Vodd(W ) (A.33)

1The naming of this group is a joke – just like SO(n) is a subgroup of O(n), Spin(n) is a subgroup of Pin(n), but in this case the ‘S’ is dropped instead of added. APPENDIX A. GROUPS, ALGEBRAS, AND REPRESENTATIONS 46

Their sum S = S+ ⊕ S− is called the spin representation and elements of S are called spinors. The half-spin representations will in turn be irreducible representations of the group Spin(n) of dimension 2n/2−1 since the dimension of W is 2n/2. In the case n = 8, the half- spin representations thus have dimension 24−1 = 8 and so are of the same dimension as the 8 fundamental representation R . In this unique situation there are automorphisms of Spin(8) or so(8) which permute the three eight dimensional representations arbitrarily – these are given by the group S3. This phenomenon is called triality. S3 is also the symmetry group of the so-called Dynkin diagram D4 of Spin(8) (see figure A.1). (Dynkin diagrams are beyond the scope of this text, but by certain rules such a diagram gives the structure of its corresponding Lie algebra or Lie group.) D4 is the only Dynkin diagram with a larger symmetry group than Z2.

Figure A.1: Dynkin diagram D4 of Spin(8).

A.4 Further Reading

For a basic introduction to group theory and abstract algebra we recommend the book by Durbin [18]. A more thorough treatment on a higher level can be found in [19] by Dummit and Foote. When it comes to Lie groups, Lie algebras and representation theory there are several useful books aimed at different audiences. For the more physically inclined reader there are the books by Georgi [20], Jones [21], Fuchs and Schweigert [22], and Cahn [23]. Here [22] is intended for graduate physics students, while [21] is more of an introductory text. [20] is written by one of the founders of Grand Unified Theories. Some sources of information aimed at readers with a main interest in mathematics are the books by Hall [24], Erdmann and Wildon [25], Humpreys [26], Jacobson [27], Fulton and Harris [28], and Baker [29]. While [24] and [25] are nice introductions for the one not familiar with the topic, [26] is a classical, but very densely written, text on the subject. [28] and [29] also contain some material on Clifford algebras and spinors. Appendix B

Principal Fiber Bundles in Particle Physics

While Appendix A introduced many of the mathematical notions necessary to understand the main text, this appendix will delve into a subject which has more the character of extra information for the interested – and mathematically inclined – reader, even though it has been found to be crucial for a deeper understanding of the foundations of particle physics and unifying theories there within. This topic, called principal fiber bundles, is from a mathemat- ical point of view the natural setting for doing particle physics, but can appear to be quite abstract for a non-mathematican. Since some of the notions show up in other areas of physics such as general relativity (which is based on semi-Riemannian manifolds), we still think it is worthwhile to include these in an appendix.1

B.1 Differentiable Manifolds

A principal fiber bundle is a special kind of differentiable or smooth manifold. A differentiable manifold is based on a topological manifold, i.e. a topological space2 M with the property that for every p ∈ M there is an open set U ⊆ M containing p and a homeomorphism n ϕ : U → ϕ(U) ⊆ R . The pair (U, ϕ) is called a chart at p. Given two non-disjoint charts −1 −1 (U1, ϕ1) and (U2, ϕ2), the overlap functions ϕ1 ◦ϕ2 and ϕ2 ◦ϕ1 are inverse homeomorphisms of the sets ϕ2(U1 ∩ U2) and ϕ1(U1 ∩ U2). If two charts are disjoint, or if the overlap functions are smooth (i.e. have continuous partial derivatives of all orders), the charts are smoothly related. An atlas for M is a family of charts {(Uα, ϕα) | α ∈ A} which are pairwise smoothly related, and such that the open sets Uα, α ∈ A, cover M. A differentiable manifold consists of a topological manifold equipped with a differentiable structure, i.e. a maximal atlas. The dimension of M is n.A diffeomorphism between two differentiable manifolds M and N is a smooth bijection with a smooth inverse – if such a function exists, M and N are diffeomorphic.

Example. One of the most important examples of a manifold in the context of particle physics is a Lie group, i.e. a differentiable manifold which is also a group such that the group struc- ture is compatible with the differentiable structure (more precisely: the group multiplication

1The content of this appendix closely follows [31]. 2Those who are not familiar with topological spaces can consult e.g. [33].

47 APPENDIX B. PRINCIPAL FIBER BUNDLES IN PARTICLE PHYSICS 48 and the function taking an element to its inverse are both smooth).

Let M be a differentiable manifold, and let C∞(M) be the vector space of smooth real- ∞ valued functions on M.A tangent vector at p ∈ M is a linear function X : C (M) → R such that X(fg) = f(p)X(g) + X(f)g(p) for all f, g ∈ C∞(M). The set of all such vectors is denoted by TpM and is called the tangent space to M at p – it has a natural vector space structure and dim TpM = dim M.A (smooth) vector field on M is a (smooth) function V on M such that V (p) = Vp ∈ TpM. The set of all smooth vector fields on M is denoted by T(M).

Example. Given a Lie group G with identity element e ∈ G, the tangent space TeG has the structure of a Lie algebra isomorphic to g = Lie(G), the Lie algebra of G.

B.2 Derivatives, Differential Forms, and Pullbacks

Given a smooth map f : M → N and a point p ∈ M, the derivative (or push-forward) of f at p ∞ is the linear function f∗p : TpM → Tf(p)N defined in the following way: f∗p(X): C (N) → R is given by f∗p(X)(g) = X(g ◦ f).

∗ A covector at p ∈ M is an element of the dual space Tp M to the tangent space TpM, i.e. a linear functional on TpM.A (real-valued) 1-form on M is a covector field on M, i.e. ∗ a function Θ on M such that Θ(p) ∈ Tp M for each p ∈ M. The set of all 1-forms on M is denoted by T∗(M).

Example. An example of a 1-form is the exterior derivative or differential df of a smooth func- ∗ tion f on M, which is defined in the following way: df(p) ∈ Tp M is given by df(p)(X) = X(f) where X ∈ TpM.

If the coordinate functions of a chart (U, ϕ) are x1, . . . , xn, i.e. ϕ(p) = (x1(p), . . . , xn(p)), ∗ 1 n a basis for Tp M in these local coordinates is {dx (p), . . . , dx (p)}.

Given a 1-form Θ on a manifold N, it can be pulled back to a 1-form F ∗Θ on a manifold ∗ M via a smooth function F : M → N. This so-called pullback is defined by (F Θ)p(X) = ΘF (p)(F∗p(X)) for each p ∈ M and each X ∈ TpM. If (U, φ) is a chart on M with coordi- nate functions x1, . . . , xn and (V, ψ) is a chart on N with coordinate functions y1, . . . , ym, i φ(U) ⊆ V and Θ = Θidy , then the expression for the pullback in these local coordinates is

∂F i F ∗Θ = (Θ ◦ F )dxj = Θ (F 1(x1, . . . , xn),...,F m(x1, . . . , xn))d(F i(x1, . . . , xn)) (B.1) ∂xj i i where ψ ◦ F ◦ φ−1(x1, . . . , xn) = (F 1(x1, . . . , xn),...,F m(x1, . . . , xn)). For g ∈ C∞(N) we get the special case F ∗(dg) = d(g ◦ F ). g can here be considered as a 0-form and we can define F ∗g = g ◦ F ∈ C∞(M). The last equality then takes the form F ∗(dg) = d(F ∗g), i.e. taking the pullback commutes with the exterior derivative on 0-forms. This is a property which can be shown to hold for any differential form.

To describe the interaction of particles in terms of principal fiber bundles, we also have to consider 2-forms on manifolds. For a smooth manifold M, a 2-form on M is a function APPENDIX B. PRINCIPAL FIBER BUNDLES IN PARTICLE PHYSICS 49

V2 Ω which for each p ∈ M assigns an element Ωp ∈ (TpM). Given a 1-form Θ on M, its exterior derivative dΘ is a 2-form on M defined in the following way: for vector fields V,W on M, dΘ(V,W ) = V (ΘW ) − W (ΘV ) − Θ([V,W ]) (B.2) where the Lie bracket of vector fields is given by [V,W ](f) = V (W (f)) − W (V (f)) for f ∈ C∞(M).

so far we have only discussed real-valued differential forms, but they can equally well take their values in a vector space V. Given a basis for V, the components of the vector-valued form will then be real-valued. Pullbacks and exterior derivatives can then be defined for vector-valued forms by performing the corresponding operations componentwise (the result can be shown to be independent of basis). A special case which is of great importance is when V is a Lie al- gebra – if the manifold under consideration is a Lie group G, this Lie algebra could be Lie(G).

Given two V-valued 1-forms ω and η on a manifold M, and a bilinear map ρ : V × V → V, their ρ-wedge product ω ∧ρ η is a V-valued 2-form defined by

(ω ∧ρ η)p(v, w) = ρ(ωp(v), ηp(w)) − ρ(ηp(v), ωp(w)) (B.3)

In the case where V has the structure of a Lie algebra g, let ρ : g × g → g be defined by ρ(A, B) = [A, B] and let [ω, η] = ω ∧ρ η. For a matrix Lie group, we can define the 0 wedge product ω ∧ η (without subscript) to be ω ∧ρ0 η where ρ (A, B) = AB. We then get [ω, η] = ω ∧ η + η ∧ ω and [ω, ω] = 2ω ∧ ω (see the Cartan Structure Equation in section B.4).

B.2.1 Left-invariant Vector Fields and the Cartan 1-form

A vector field X on a Lie group G is said to be left-invariant if (Lg)∗h(X(h)) = X(gh) for all g, h ∈ G. The set of all left-invariant vector fields on G is (isomorphic to) its Lie algebra g.

∗ A 1-form Θ on G is left-invariant if Θ(h) = (Lg) (Θ(gh)) for all g, h ∈ G, or equivalently ∗ Θ(g) = (Lg−1 ) (Θ(id)) for all g ∈ G.

A left-invariant g-valued 1-form on G is the Cartan (canonical) 1-form, defined in the follow- ing way: Θg = Θ(g): TgG → g = TidG is given by

Θg(v) = (Lg−1 )∗g(v)

1 n If {e1, . . . , en} is a basis for g and {Θ ,..., Θ } is the unique left-invariant real-valued 1-forms 1 n on G for which {Θ (id),..., Θ (id)} is the basis dual to {e1, . . . , en}, then the Cartan 1-form 1 n on G is given by Θ = Θ e1 + ··· + Θ en. The Maurer-Cartan equations for Θ assert that 1 dΘk = − Ck Θi ∧ Θj, k = 1, . . . , n 2 ij

k where Cij are the structure constants for g relative to {e1, . . . , en}.

−1 For G = GL(n, R), in local coordinates the Cartan 1-form can be written as Θ(g) = g dx(g). APPENDIX B. PRINCIPAL FIBER BUNDLES IN PARTICLE PHYSICS 50

B.2.2 Killing Form and Riemannian Metric

If G is connected and semisimple, then the Killing form κ : g × g → R is negative definite iff G is compact. In that case, hA, Bi = −κ(A, B) is a positive definite inner product on g. This gives rise to a bi-invariant Riemannian metric g on G defined by

gg(v, w) = h(Lg−1 )∗g(v), (Lg−1 )∗g(w)i

for g ∈ G and v, w ∈ TgG.

B.3 Principal Fiber Bundles

Let M be a smooth manifold and G a Lie group. A smooth principal fiber bundle P over M with structure group G, written G,→ P →π M, consists of a differentiable manifold P , a smooth map π : P → M, and a smooth right action p.g of G on P such that the following holds: • π(p.g) = π(p) for all p ∈ P and all g ∈ G

• For each x0 ∈ M there exists an open set U in M containing x0 and a diffeomorphism Ψ: π−1(U) → U × G of the form Ψ(p) = (π(p), ψ(p)) where ψ : π−1(U) → G satisfies ψ(p.g) = ψ(p)g for all p ∈ π−1(U) and all g ∈ G. The first condition says that the action preserves fibers. In the second condition, (U, Ψ) is called a local trivialization.

Given a smooth principal bundle G,→ P →π M, the fibers π−1(x), x ∈ M, are diffeomorphic to G, so for each p ∈ P , the tangent space TpP contains a subspace isomorphic to g, called the vertical subspace of TpP , and is denoted by Vertp(P ). An isomorphism g → Vertp(P ) is given by A 7→ A#(p), where the vector field A# is defined by d A#(p) = (σ ) (A) = (p. exp(tA))| p ∗id dt t=0 # where σp : G → P is defined by σp(g) = p.g. The vector field A is called the fundamental vector field on P determined by A (thought of as a matrix).

B.4 Connections, Curvature, and Gauge Fields

A connection form on a principal bundle G,→ P →π M is a smooth g-valued 1-form ω on P which satisfies ∗ • (σg) ω = adg−1 ◦ ω for all g ∈ G, i.e. −1 ωp((σp)∗p.g−1 (v)) = g ωp.g−1 (v)g

for all g ∈ G, p ∈ P and v ∈ Tp.g−1 P . APPENDIX B. PRINCIPAL FIBER BUNDLES IN PARTICLE PHYSICS 51

• ω(A#) = A for all A ∈ g, i.e. # ωp(A (p)) = A for all A ∈ g and p ∈ P . A local cross-section s : U → π−1(U) of the bundle is called a local gauge and the pullback A = s∗ω of ω to U ⊆ M by s is called a local gauge potential (in gauge s).

−1 Let {(Uj, Ψj)}j∈J be a trivializing cover of M and sj : Uj → π (Uj) a cross-section as- ∗ sociated with (Uj, Ψj). Then the family {Aj = sj ω}j∈J of local gauge potentials satisfies ∗ Aj = ad −1 ◦ Ai + gijΘ gij

for all i, j ∈ J with Ui ∩ Uj 6= ∅ where gij : Ui ∩ Uj → G are the transition functions. When ∗ G is a matrix group, the g-valued 1-forms gijΘ on Ui ∩ Uj are given by ∗ −1 (gijΘ)x(v) = (gij(x)) dgij(x)(v)

where dgij is the entrywise differential of gij : Ui ∩ Uj → G. This gives −1 −1 Aj = gij Aigij + gij dgij Given a connection form ω on P , for each p ∈ P , the horizontal subspace is defined by Horp(P ) = {v ∈ TpP : ωp(v) = 0}. Then

TpP = Horp(P ) ⊕ Vertp(P ) H V H so each v ∈ TpP can be written uniquely as v = v + v where v ∈ Horp(P ) and V v ∈ Vertp(P ).

Let ω be a connection form on G,→ P →π M. Its curvature Ω is defined by H H Ωp(v, w) = (dω)p(v , w ) and is a g-valued 2-form globally defined on P . The Cartan Structure Equation asserts that 1 Ω = dω + [ω, ω] = dω + ω ∧ ω 2 The curvature Ω of a connection ω on G,→ P →π M is called a gauge field on G,→ P →π M. The pullback s∗Ω of the curvature by a local cross-section s : U → π−1(U) is called the local field strength (in gauge s), and is denoted by F. With A = s∗ω and F = s∗Ω the Cartan Structure Equation becomes F = dA + A ∧ A When the domain U of s is a coordinate neighborhood with coordinate functions x1, . . . , xn α 1 α β we can write A = Aαdx and F = 2 Fαβdx ∧ dx where Aα and Fαβ are g-valued functions on U. Then Fαβ = ∂αAβ − ∂βAα + [Aα, Aβ] The relation between the field strength in different gauges is −1 Fj = gij Figij

In the case G = U(1), the Lie algebra g = u(1) is abelian, so Fj = Fi and we get a globally defined field strength 2-form F. APPENDIX B. PRINCIPAL FIBER BUNDLES IN PARTICLE PHYSICS 52

B.5 Matter Fields

While a gauge field is connected to a principal bundle, a matter field will be connected to a vec- tor bundle associated with a principal bundle, i.e. a bundle where the fibers are vector spaces.

Let ρ : G → GL(V) be a smooth representation of G on the vector space V. Then g.v = (ρ(g))(v) is a smooth left action of G on V. Given a principal bundle G,→ P →π M, we can πρ define an associated vector bundle G,→ P ×ρ V → M in the following way: • a right-action of G on P × V is given by (p, v).g = (p.g, g−1.v)

• P ×ρ V = {[p, v]:(p, v) ∈ P × V} is the orbit space under this action

• πρ : P ×ρ V → M is defined by πρ([p, v]) = π(p) −1 −1 For any x ∈ M, the fiber over x, i.e. πρ (x) = {[p, v]: v ∈ V}, where p is any point in π (x), is a copy of V.

The adjoint bundle of G,→ P →π M, where G is a matrix group, and Ad : G → GL(g) −1 π is given by Adg(A) = gAg , is the vector bundle associated with G,→ P → M by Ad, and is denoted by Ad P = P ×Ad g. The fibers are copies of the Lie algebra g.

If U ⊆ M is open, then a smooth map φ : π−1(U) → V is equivariant (w.r.t. the actions of G on P and V) if φ(p.g) = g−1.φ(p) for all p ∈ π−1(U) and g ∈ G. There is a bijective correspondence between equivariant maps and local cross-sections of the associated vector bundle.

A particle coupled to a gauge field is described by a V-valued wave function ψ defined lo- cally on M. ψ depends on the gauge potentials A = s∗ω, so when the gauge is changed, so is the wave function. However, on P ×ρ V the wave functions piece together to a glob- ally defined cross-section, which corresponds to a globally defined equivariant map φ : P → V.

Let G,→ P →π M be a principal G-bundle and ρ : G → GL(V) a representation of G on V. An equivariant V-valued map φ : P → V is called a matter field (of type ρ) on G,→ P →π M. If V = g is the adjoint representation, φ is called a Higgs field.

Let ω be a connection form and φ a matter field on G,→ P →π M. Then the covariant exterior derivative dωφ of φ is defined by ω H (d φ)p(v) = (dφ)p(v ) This is a V-valued 1-form on P satisfying ∗ ω −1 ω σg (d φ) = g .d φ n for each g ∈ G. Now let g act on V: if g consists of n×n matrices and V = R , then A.v = Av is the natural action, and if V = g, then A.B = [A, B] is the adjoint action. By using this action, define a V-valued 1-form ω.φ on P by

(ω.φ)p(v) = ωp(v).φ(p) APPENDIX B. PRINCIPAL FIBER BUNDLES IN PARTICLE PHYSICS 53

for each p ∈ P and v ∈ TpP . A computational formula analogous to the Cartan Structure Equation for Ω is then given by dωφ = dφ + ω.φ

B.6 General Framework for Classical Gauge Theories

The basic mathematical ingredients required to describe (at the classical level) the interaction of a particle with a gauge field are the following:

1,3 4 4 1. A smooth, oriented (semi-)Riemannian manifold M, e.g. R or S = R ∪ {∞}. 2. A finite dimensional inner product space (V, h , i) (the internal space) where the wave function of a particle takes its values.

3. A matrix Lie group G and a representation ρ : G → GL(V) such that

hρ(g)(v), ρ(g)(w)i = hv, wi

4. A smooth principal G-bundle P over M: G,→ P →π M

5. A connection ω on G,→ P →π M with curvature Ω (a gauge field on P ). (ω is a g-valued 1-form on P , and Ω is a g-valued 2-form on P .) If s : U → P is a local cross-section (local gauge), then A = s∗ω is the local gauge potential and F = s∗Ω is the local field strength.

6.A matter field, which can be described in two equivalent ways:

• A map φ : P → V which is equivariant (w.r.t. ρ): φ(p.g) = g−1.φ(p) (In the case of a Higgs field, φ is g-valued.)

πρ • A global cross-section of the vector bundle G,→ P ×ρ V → M.

7. A smooth potential function V : V → R≥0 such that V (g.v) = V (v). V ◦ φ : P → R≥0 1 2 describes the self-interaction energy of the matter field φ, e.g V ◦ φ = 2 mkφk . 8. An (energy) functional A(ω, φ), the stationary points of which are the physically signif- icant field configurations (ω, φ). The functional is typically of the form Z 2 ω 2
A(ω, φ) = c kFωk + c1kd φk + c2V ◦ φ M The (local) minima must satisfy the Euler-Lagrange equations. These are the appropri- ate field equations (the “equations of motion”) for the gauge theory. In physics, usually only finite action configurations (ω, φ) are interesting, i.e. those for which A(ω, φ) < ∞. This is the case if M is compact, or if there is appropriate asymptotic behavior for the terms in the integrand. APPENDIX B. PRINCIPAL FIBER BUNDLES IN PARTICLE PHYSICS 54

B.7 Yang-Mills Theory

B.7.1 Pure Yang-Mills Theory

When c1 = c2 = 0, so there are no matter fields present, A only depends on ω, and it is called the Yang-Mills action and is written Z 2 YM(ω) = c kFωk M The Euler-Lagrange equations for YM are called the Yang-Mills equations and can be written ω ∗ d Fω = 0 ∗ ω where Fω is the Hodge dual of the Ad P -valued 2-form Fω on M, and d is the covariant exterior derivative. Independent of the action, the field Fω also satisfies a purely geometrical constraint known as the Bianchi identity: ω d Fω = 0

B.7.2 Pure Yang-Mills Electromagnetic Field Theory 1,3 In the theory of classical electrodynamics, M is the open submanifold of R given by re- moving the worldlines of the charges creating the electromagnetic field. The gauge group is G = U(1) with the abelian Lie algebra g = u(1) = Im C. We are considering a principal U(1)-bundle U(1) ,→ P →π M with a connection ω (and no matter field). Since u(1) is abelian, Ω = dω. The local field strength is given by 1 F = − iF dxα ∧ dxβ 2 αβ where  0 −E1 −E2 −E3   E1 0 B3 −B2  (Fαβ) =    E2 −B3 0 B1  E3 B2 −B1 0 1 α β Two scalar invariants associated with F = 2 Fαβ dx ∧ dx are 1 αβ ~ 2 ~ 2 2 FαβF = kBk − kEk

1 ∗ αβ ~ ~ 4 Fαβ F = E · B The Yang-Mills action is Z 1 αβ 0 1 2 3 YM(ω) = − FαβF dx dx dx dx M 4 and the corresponding Yang-Mills equations are d ∗F = 0 while the Bianchi identity is dF = 0 These are the source free Maxwell equations. APPENDIX B. PRINCIPAL FIBER BUNDLES IN PARTICLE PHYSICS 55

B.8 Further Reading

The main source of information for this appendix is [31] by Naber. This book contains a very brief introduction to differentiable manifolds and also covers some topics in topology such as de Rham cohomology and characteristic classes. For the reader who wants a more thorough introduction to differential geometry the prequel [30] to [31] can be recommended, as well as the excellent book by Lee [34]. While [30, 31] contain some physical motivation for the study of geometry and topology, [34] is more directed towards the mathematically inclined reader. A book which is also of a very mathematical taste but which still covers principal fiber bundles as a means for doing particle physics is [32]. Another possible source of information on this topic is [35]. Bibliography

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