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.