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Aspects of Neutrino Physics and CP Violation Aspects of Neutrino Physics and CP Violation Pedro Miguel Lopes Loreto dos Santos Disserta¸c~aopara a obten¸c~aode Grau de Mestre em Engenharia F´ısicaTecnol´ogica J´uri Presidente: Prof. Maria Teresa Haderer de la Pe~naStadler Orientador: Dr. Maria Margarida Nesbitt Rebelo da Silva Vogal: Dr. David Emmanuel-Costa Vogal: Dr. Ivo de Medeiros Varzielas Outubro 2011 Acknowledgements I dedicate this thesis to my family. To Neli Maria Pereira Lopes. To Jo~aoAlberto Loreto dos Santos. To Av´oCarlota. To Av´oConcei¸c~ao. To Av^oChico Lopes. To Av^oJoaquim. i Resumo Este estudo discute as propriedades f´ısicasdos neutrinos e do sector lept´onico.Nesse sentido, ´eanalisada n~ao s´oa sua natureza e correspondente estrutura da matriz de mistura lept´onica,como tamb´ems~aoapresentados diferentes tipos de mecanismos de seesaw associados a um termo de massa de Majorana. Fazemos uma breve revis~aoda oscila¸c~aode neutrinos e do duplo decaimento beta sem neutrinos, apresentando o estado observacional e experimental actual, assim como os ´ultimosresultados relativos ao ^angulode mistura θ13 obtidos pelo T2K. S~ao,de seguida, explicadas as condi¸c~oesnecess´ariase suficientes para a produ¸c~aode uma assimetria bari´onica,ou mais especificamente as condi¸c~oesde Sakharov, e exploramos uma delas, a viola¸c~ao CP, no sector lept´onico.Finalmente focamos a nossa aten¸c~aona cria¸c~aodessa assimetria pela leptog´enese, mostrando de uma forma qualitativa que esta cont´emtodos os elementos necess´arios`asua gera¸c~ao. Palavras-chave: f´ısicade neutrinos, part´ıculasde Majorana, viola¸c~aoCP, leptog´enese. iii Abstract A concise analysis on the physical properties of neutrinos and the leptonic sector is presented. In this context, we discuss its nature (Dirac vs. Majorana) and corresponding structure of the leptonic mixing matrix, along with the different types of seesaw mechanism. We briefly review neutrino oscillation and double-β decay scheme, presenting the current observational and experimental status, including the novel results from T2K regarding θ13. After explaining the theoretical conditions for producing a matter-antimatter asymmetry in the Universe, or more specifically the Sakharov's conditions, we explore one of them, CP violation, in the leptonic sector. Finally, we focus our attention in the creation of that asymmetry through leptogenesis, and we show that, in this scenario, all the qualitative ingredients are guaranteed once the seesaw mechanism is assumed to be the source of neutrino masses. Keywords: neutrino physics, Majorana particles, CP violation, leptogenesis. v Contents Acknowledgements . i Resumo . iii Abstract . v List of Tables . ix List of Figures . xi 1 Introduction 1 2 Theoretical background 3 2.1 Majorana Spinor . 3 2.1.1 Dirac Equation . 3 2.1.2 Charge conjugation . 4 2.1.3 Lorentz transformation . 5 2.1.4 Dirac and Majorana mass terms . 6 2.1.5 A few words on Majorana mass term . 7 2.2 Discrete transformations . 8 2.2.1 Parity P . 9 2.2.2 Charge Conjugation C . 10 2.2.3 CP . 11 2.2.4 Photon Field . 12 2.2.5 Time Reversal T . 12 2.2.6 CPT Theorem . 13 3 Experimental situation 15 3.1 Reconstructing mass matrices from experiments . 17 3.2 Neutrino oscillations experiments . 19 4 Minimal extension of the Standard Model of particle physics 23 4.1 Extend Majorana masses to the SM . 24 4.2 Additional relevant CP transformations . 26 4.3 Seesaw mechanism . 27 4.3.1 Type I seesaw mechanism . 27 4.3.2 Type II seesaw mechanism . 30 4.3.3 A final word on seesaw mechanism . 32 4.4 Leptonic mixing matrix in low energy physics . 33 4.4.1 Parametrisation . 36 vii 4.4.2 Degenerate neutrino mass matrix . 39 4.4.3 Rephasing invariance . 40 4.4.4 Non-Unitarity . 41 4.5 Neutrino Oscillations . 41 4.6 Neutrinoless double beta-decay . 44 4.7 Baryon Asymmetry of the Universe and Sakharov's Conditions . 46 5 CP violation in the leptonic sector 49 5.1 General CP conditions . 49 5.2 CP violating phases . 49 5.3 CP Parities . 52 5.4 CP violating phases in seesaw mechanism type I . 55 5.5 Leptonic unitary triangles . 56 5.6 Low energy invariants in leptonic sector . 58 5.7 Leptogenesis related invariants . 63 6 Leptogenesis 65 6.1 CP violation ()............................................ 67 6.1.1 A further look on CP violation . 69 6.2 Out-of-equilibrium dynamics (η) .................................. 70 6.3 Lepton and B + L violation (Csphal)................................ 72 6.4 Baryon asymmetry from leptogenesis . 76 7 Conclusions 77 A Some properties of the matrix C 79 B General mass term 81 C The third low energy invariant 83 D Calculating CP violation in Leptogenesis 85 D.1 The baryon asymmetry as an initial condition . 85 D.2 Implication of unitarity and CPT in decays . 86 D.3 Hubble expansion rate . 87 D.4 Decay rate minimum . 87 D.5 Thermal equilibrium and the third Sakharov condition . 88 viii List of Tables 3.1 Neutrino parameters summary. 17 2 3.2 sin 2θ13 experimental results summary. 17 3.3 Five main Long-baseline neutrino oscillation experiments. 20 5.1 Number of CP violating phases. 51 D.1 SU(2) U(1) quantum numbers. 89 × ix List of Figures 2.1 Difference between massive Dirac and Majorana particles. 8 3.1 Neutrino mass patterns as indicated by the current data. 16 4.1 Neutrino flavour oscillation in vacuum. 42 4.2 Double-β transition. 44 4.3 0νββ decay process. 45 4.4 Schematic departure from equilibrium of the B violating process X Y + B. 48 ! 6.1 The diagrams contributing to the CP asymmetry αα. ...................... 67 6.2 Sphaleron process. 74 xi Chapter 1 Introduction The historic discovery of neutrino oscillations marks a turning point in particle and nuclear physics and implies that neutrinos have mass. This possibility was first suggested in late seventies, where theory tried to explain the smallness of neutrino mass through seesaw mechanism [2{5]. Due to the successfulness of the Standard Model (SM) on explaining most of the high energy experi- ments, we are induced to extend it in a minimal way so that we describe neutrino oscillation and explore different possibilities. One of the most interesting relies on the nature of neutrinos - they are either Dirac particles possessing distinctive antiparticles, or Majorana, neutral particles which are truly identical to their antiparticles [6]. On the one hand, massive Dirac neutrinos are realised in gauge theories in which the lepton charges, Le, Lµ and Lτ , are not conserved, but a specific combination of the latter, e.g. L = Le + Lµ + Lτ , is conserved; on the other hand, massive Majorana neutrinos arise if no lepton charge is conserved by the electroweak interactions. Thus, the question of the nature of massive neutrinos is directly related to the question of the fundamental symmetries of the elementary particle interactions. It is important to keep in mind that different phenomena arise depending on the nature of neutrinos. When we take neutrinos to be Dirac particles, oscillation has been our source of experimental results [7,8]. However, processes such as double-β decay are viable in the case neutrinos are Majorana, and their observation can not only indeed confirm it, but can also provide constraints on the general mass scale of neutrinos, to which we only have an upper bound. From the problem of the nature of massive neutrinos also arise different possibilities on the way CP is violated. It is unambiguously established that in the quark sector there is one CP violating phase [9]. In the context of Dirac neutrinos one would possibly have an analogous CP violating phase in the leptonic sector. There is, however, the possibility that instead of one CP violating phase, we have three CP violating phases [10] in the decoupling limit, in the case neutrinos are Majorana. This study addresses a rigorous analysis to this subject. In the context of CP violation there is the puzzle of the baryon asymmetry of the Universe (BAU). We may wonder why such asymmetry is not an initial condition. Indeed, it would require not only a very precise fine-tuning, but we would also have such asymmetry exponentially diluted away by inflation. The Standard Model contains all the ingredients proposed by Sakharov [11] to dynamically generate the observed BAU, but, yet, it fails to explain an asymmetry as large as the one observed (see, e.g. [12]), and for that reason additional sources of CP violation are called for. This new physics must, first, distinguish matter from antimatter in a more pronounced way than the weak interactions of SM do. Second, it should provide a departure from thermal equilibrium during the history of the universe, or modify the electroweak phase transition. It that 1 context we present leptogenesis [13], which provides a very elegant explanation regarding this issue. This study is organized as follows. In chapter 2, we present some theoretical formalism involving neutrinos, especially regarding the Majorana spinor and the CP transformations we shall apply later on in this study. In chapter 3 we list the current experimental bounds on the mixing angles and mass-squared differences, and present a brief description about how different parameters can be (are) experimentally determined. In chapter 4 we do not only present our framework, including our minimal extension to the Standard Model and the different seesaw mechanisms, but we also briefly explore the two most relevant phenomena involving neutrinos: neutrino oscillation and double-β decay. To finalise this general physics part of the study, we also present the Sakharov's conditions. In chapter 5 we proceed within our framework with a rigorous analysis of CP violation in the leptonic sector.
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