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Neutrinos, Symmetries and the Origin of Matter Engineering Physics Neutrinos, Symmetries and the Origin of Matter João Tiago Neves Penedo Thesis to obtain the Master of Science Degree in Engineering Physics Examination Committee Chairperson: Prof.a Doutora Maria Teresa Haderer de la Peña Stadler Supervisor: Prof. Doutor Filipe Rafael Joaquim Members of the Comittee: Prof. Doutor Gustavo da Fonseca Castelo Branco Prof. Doutor Ricardo Jorge Gonzalez Felipe November 2013 \Each piece, or part, of the whole of nature is always merely an approximation to the complete truth, or the complete truth so far as we know it. In fact, everything we know is only some kind of approximation, because we know that we do not know all the laws as yet. Therefore, things must be learned only to be unlearned again or, more likely, to be corrected." { Richard P. Feynman (1918-1988) The Feynman Lectures on Physics, Vol. I i ii Acknowledgements This thesis is not a work of mine alone: behind the stage curtain, a larger cast hides. To them I thank for not only helping me construct a symmetric work, in the Vitruvian sense, but also for keeping me sane in the process (well, as sane as possible at least). I would like to start by thanking my supervisor, Professor Filipe Joaquim, for his guidance, patience, and incessant encouragement. Being one of the few great professors I had, he has been responsible for introducing me to the world of particle physics research, the do's and don'ts of the field, and for providing all the assistance needed in solving problems and answering questions, big or small, without exception. I am additionally indebted to Funda¸c~aopara a Ci^enciae Tecnologia (FCT) and Centro de F´ısica Te´oricade Part´ıculas(CFTP), thanking, in particular, the kindness of Cl´audiaRom~ao,who has lent a helping hand whenever needed, and the support of Professor Jorge Rom~ao,who worked towards my participation in the ICTP Summer School on Particle Physics, earlier this year. There, I was given the chance to learn from the leading experts in the field, and would like to express my gratitude towards Doctor Alejandro Ibarra for a most helpful lightning-discussion as well as a saving reference. I would also like to thank my family for their support and encouragement, and for enabling my crazy endeavors, without understanding, most of the time, what the heck I am doing. I thank my friends for the chaotic yet enjoyable sequence of events one calls a physics course. A special mention must be made to Pedro Boavida and Ant´onioCoutinho, to whom I thank for our spontaneous (and crucial) discussions about physics, Copernicus and chameleons. I end by expressing my thanks to Jo~aoLoureiro, who was there even if two thousand kilometers away, and to S´ılvia Conde, for a superposition of all possible reasons. iii iv Este trabalho foi financiado pela Funda¸c~aopara a Ci^enciae Tecnologia, sob o projecto PTDC/FIS/102120/2008. This work was supported by Funda¸c~aopara a Ci^enciae Tecnologia, under the grant PTDC/FIS/102120/2008. v vi Resumo As simetrias como leis de invari^anciadesempenham um papel fundamental na constru¸c~aode teo- rias f´ısicas. Em particular, as simetrias de gauge est~aona base do presente conhecimento do mundo subat´omico,que assenta no Modelo Padr~aoda f´ısicade part´ıculas. Apesar de repetido sucesso, este modelo tem que ser necessariamente expandido `aluz da exist^enciade massas e mistura de neutrinos. Na presente disserta¸c~aos~aoexploradas extens~oesdo Modelo Padr~aobaseadas no mecanismo seesaw onde a supress~aoda massa dos neutrinos ´enaturalmente explicada. Massas de neutrinos n~aonulas conduzem a mistura lept´onica,cuja estrutura se aproxima a um padr~ao tribimaximal, apontando para a poss´ıvel presen¸cade simetrias discretas na teoria a altas energias { como a invari^anciasob transforma¸c~oes do grupo A4, considerado neste trabalho. O Modelo Padr~aorevela-se igualmente insuficiente na explica¸c~aoda assimetria bari´onicado Universo. Nos modelos seesaw ´eposs´ıvel gerar dinamicamente essa assimetria atrav´esdos decaimentos dos novos estados pesados (fora de equil´ıbriot´ermico)mediante o mecanismo de leptog´enese,cuja efici^encia´ede- terminada numericamente resolvendo o sistema de equa¸c~oesde Boltzmann adequado. Nesta disserta¸c~ao, apresenta-se a an´alisede um modelo particular para viola¸c~aoespont^aneada simetria CP onde se explicam as massas e mistura de neutrinos impondo uma simetria discreta A4. A implementa¸c~aodo mecanismo de leptog´eneseneste contexto ´ediscutida em detalhe. Palavras-chave: Assimetria bari´onicado Universo; Leptog´enese;Massa e mistura de neutrinos; Mecanismo seesaw; Simetrias; Viola¸c~aode CP vii viii Abstract Symmetries, understood as laws of invariance, play a fundamental role in the development of physics. In particular, gauge symmetries are at heart of our current understanding of the subatomic world, which relies on the Standard Model of particle physics. Despite its repeated successes, this model must neces- sarily be extended to accommodate the experimental observation of nonzero neutrino masses and mixing. In this thesis, we explore seesaw extensions of the Standard Model, where heavy states mediate neutrino mass generation and the smallness of these masses is naturally accounted for. Nonvanishing neutrino masses allow for leptonic mixing, whose structure strongly differs from that of quark mixing. The closeness of the lepton mixing matrix to the tribimaximal pattern points to the presence of discrete symmetries in the underlying high-energy theory { such as invariance under transformations of the A4 group, considered in this work. The Standard Model also fails to provide a satisfactory mechanism for the generation of the baryon asymmetry of the Universe. A remarkable feature of the seesaw extensions is the possibility that the out-of-equilibrium decays of the new heavy states are responsible for the dynamical generation of this asymmetry. This corresponds to the leptogenesis mechanism, whose efficiency is here determined by numerically solving a system of Boltzmann equations. Additionally, a particular model for spontaneous leptonic CP violation is analysed where neutrino masses and mixing are explained imposing an A4 discrete symmetry. The implementation of the leptogenesis mechanism in this context is discussed in detail. Keywords: Baryon asymmetry of the Universe; CP violation; Leptogenesis; Neu- trino masses and mixing; Seesaw mechanism; Symmetries ix x Contents Acknowledgements iii Resumo vii Abstract ix List of Figures xiv List of Tables xv List of Abbreviations xvii 1 Symmetries and Asymmetries in Nature 1 1.1 Evolution of the Concept of Symmetry . .3 1.2 Groups and Symmetry . .5 1.3 Symmetry in Physics . .8 1.3.1 From Classical to Quantum Mechanics . .8 1.3.2 New Kinds of Symmetry . .9 1.3.3 The Discrete Symmetries C, P and T . 10 1.3.4 Symmetry Breaking . 12 1.4 A Philosophical Interlude . 14 1.5 The Asymmetry of Existence . 15 1.5.1 Experimental Evidence . 15 1.5.2 The Tuning of Initial Conditions . 15 1.5.3 The Possibility of a B-Symmetric Universe . 16 2 The Standard Model of Particle Physics and (slightly) Beyond 17 2.1 Recap of the Electroweak Sector of the SM . 17 2.1.1 Neutral and Charged Electroweak Currents . 19 2.1.2 The Higgs Mechanism . 20 2.1.3 Fermion Masses and Mixing . 22 2.2 Neutrinos Beyond the SM . 25 2.2.1 The Neutrino Mass Term . 25 2.2.2 The Seesaw Mechanism . 27 xi 3 Lepton Mixing and Discrete Family Symmetries 35 3.1 Lepton Mixing . 35 3.2 Discrete Family Symmetries . 39 3.2.1 Symmetries of the Mass Matrices . 39 3.2.2 Direct vs. Indirect Models . 40 3.3 An A4 Model with Spontaneous CP Violation . 41 3.3.1 Spontaneous CP Violation . 44 3.3.2 Neutrino Masses and Mixing . 45 3.3.3 Nonzero Reactor Neutrino Mixing Angle . 47 4 Baryogenesis through Leptogenesis 49 4.1 Topics of Cosmology and Thermodynamics . 49 4.1.1 Cosmological Inflation . 49 4.1.2 Equilibrium Thermodynamics . 50 4.1.3 Expansion, Entropy and Degrees of Freedom . 52 4.1.4 Brief Thermal History of the Universe . 53 4.2 The Sakharov Conditions . 54 4.3 Is the SM Enough? . 56 4.4 Thermal Leptogenesis . 59 4.4.1 CPT, Unitarity and CP Asymmetries . 60 4.5 Boltzmann Equation(s) . 62 4.5.1 Two-body Decays and Inverse Decays . 63 4.5.2 2 2 Scatterings . 65 $ 5 Type II Seesaw Leptogenesis 69 5.1 Flavoured CP Asymmetries from Triplet Decays . 69 5.2 Boltzmann Equations for Type II Seesaw . 71 5.3 Scattering Reaction Densities . 74 5.4 Leptogenesis in an A4 Model . 75 Conclusions 79 Bibliography 88 A Computing Diagrams with Majorana Fermions 89 B Clebsch-Gordan Coefficients for A4 91 B.1 General Description of the Group . 91 B.2 Choice of an Explicit 3D Representation . 92 B.3 The Tensor Product Representation . 93 B.4 Computing the CGCs . 94 xii List of Figures 1.1 Selected symmetry drawings of M. C. Escher (1941) . .1 1.2 Results of subjecting an artificial \quasi-lattice" based on a Penrose tiling to optical diffraction (left), obtained by A. Mackay in 1982, and its physical analogue (right): elec- tron diffraction patterns of an aluminium-based icosahedral quasicrystal, published by D. Shechtman et al. in 1984 . .2 2.1 Mass hierarchy of the elementary fermions observed in Nature. Mass values and uncer- tainties are obtained from J. Beringer et al. (Particle Data Group) 2012 and references therein (light quarks present the highest relative mass uncertainties) . 25 2.2 Exchange interactions which in the effective theory give rise to the Weinberg operator of (2.45). Seesaw types I and III correspond to the exchange of fermion fields NR and ΣR, respectively (left diagram), while the type II seesaw mechanism is implemented through the exchange of scalar fields ∆ (right diagram) .
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