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Discrete Family Symmetries and Tri-Bimaximal Leptonic Mixing

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Discrete Family Symmetries and Tri-Bimaximal Leptonic Mixing Discrete Family Symmetries and Tri-Bimaximal Leptonic Mixing Renato Miguel Sousa da Fonseca Dissertação para a obtenção de Grau de Mestre em Engenharia Física Tecnológica Júri Presidente: Gustavo Fonseca Castelo-Branco Orientador: Jorge Manuel Rodrigues Crispim Romão Co-orientador: Joaquim Inácio da Silva Marcos Vogais: David Emanuel da Costa Setembro 2008 ii Acknowledgements I am grateful to CFTP members and in particular to my supervisor, co-supervisor, and David Emmanuel Costa for the help given. The support of my colleagues and family was also of the utmost importance to me. iii iv Resumo Recentemente foi descoberto que os neutrinos oscilam pelo que têm massa. Contrariamente ao que se passa com os quarks, os ângulos de mistura dos leptões são grandes. De facto a mistura leptónica aproxima-se dum limite conhecido por tri-bimaximal mixing. O Modelo Padrão da física de partículas tem tido um enorme sucesso a descrever o comportamento da Natureza a um nível microscópico e pode facil- mente ser estendido de forma a ter em conta estes novos factos. No entanto, muitos dos seus parâmetros não se encontram constrangidos do ponto de vista teórico. Uma forma de introduzir relações em alguns dos seus parâmetros - massas e matrizes de mistura dos fermiões - consiste em postular a existência de simetrias discretas de sabor na teoria. Numa primeira parte deste trabalho, são revistas as bases da física teórica de partículas. Segue-se o estudo de uma classe de extensões do Modelo Padrão que contêm simetrias discretas de sabor, neutrinos com massa e um ou mais dubletos de Higgs. O objectivo é deduzir propriedade genéricas destes modelos e daí extrair conclusões acerca da possibilidade de se ter tri-bimaximal mixing leptónico. Concluo que tal não é possível para a classe de modelos considerados com apenas um dubleto de Higgs invariante. Palavras-chave: Extensões do Modelo Padrão, Mistura Leptónica, Simetrias Discretas, Sime- trias Discretas de Família, Simetrias Discretas de Sabor, Tri-Bimaximal Mixing v vi Abstract It was recently discovered that neutrinos oscillate and therefore they must have mass. In contrast with the quark sector, leptonic mixing angles are large. In fact leptonic mixing is said to be approximately tri-bimaximal. The Standard Model of particle physics has been very successful at describing Nature at the microscopic level and it can easily be extended to take into account these new facts. However, there are no theoretical constraints on many of its parameters. One way to introduce relations between some of its parameters - the fermions’ masses and mixing matrices - is to postulate the existence of discrete family symmetries in the theory. In the first part of this work, the basics of particle physics theory are reviewed. This is followed by the study of a class of extensions of the Standard Model which contain discrete family symmetries, massive neutrinos, and one or more Higgs doublets. The aim is to deduce general properties of these models and find out if they can lead to leptonic tri-bimaximal mixing. I conclude that for the class of models considered, with a single invariant Higgs doublet, it is impossible to obtain such a mixing. Keywords: Discrete Symmetries, Discrete Family Symmetries, Discrete Flavour Symmetries, Leptonic Mixing, Extensions of the Standard Model, Tri-Bimaximal Mixing vii viii Contents Acknowledgements . iii Resumo . v Abstract . vii List of Tables . xiii List of Figures . xv Abbreviations . xvi I The Standard Model 1 1 Generalities 3 1.1 Field content . 3 1.2 Dynamics of the fields . 4 1.3 Gauge invariance . 7 1.3.1 Noether Theorem - Symmetry and conservation laws . 8 1.3.2 Gauge theories . 9 1.3.3 Spontaneous symmetry breaking (SSB) . 11 1.3.4 The Higgs mechanism . 14 1.4 Renormalization . 15 2 The Standard Model 17 2.1 Quantum Chromodynamics . 17 2.2 Electroweak theory . 20 2.2.1 A bit of history . 20 2.2.2 The Glashow-Weinberg-Salam (GWS) model . 21 II Mass and Mixing Matrices 27 3 Quarks 31 3.1 CKM matrix parametrizations . 32 3.2 Unitarity constraints on the CKM matrix . 32 3.3 CP violating phase . 33 3.4 Vckm’s experimental values . 33 4 Leptons 35 4.1 Accounting for neutrinos’ mass . 35 4.2 The seesaw mechanism . 35 4.3 Neutrino oscillations . 38 4.4 Experimental values of the neutrinos’ masses and mixing . 39 4.5 Tri-bimaximal mixing . 41 III Models in literature with tri-bimaximal mixing 43 5 Frequently used groups 47 5.1 Cn ................................................. 47 5.2 Sn ................................................. 47 ix 5.3 An ................................................. 48 5.4 ∆ 3n2 .............................................. 48 6 Models with tri-bimaximal mixing 51 IV Theoretical considerations on models based on discrete family symme- tries 57 7 Models with one invariant Higgs doublet 61 7.1 General Considerations . 61 7.2 Some Definitions . 61 7.3 Establishing the different scenarios . 62 7.4 Relevant Lagrangian Mass Terms . 62 7.5 Synchronized action of the groups . 63 7.6 Independent action of the groups . 66 7.7 More than one group acting on each multiplet . 67 7.8 Summary . 69 7.9 Relation between an irrep and its complex conjugate . 70 7.10 Can models with one Higgs doublet accommodate experimental data? . 75 8 Models with more than one Higgs doublet 79 8.1 Paradigm of the analysis . 79 8.2 Properties of the Mi matrices . 81 8.3 Linear combinations of Mis................................... 89 8.4 Is tri-bimaximal mixing possible? . 91 8.4.1 An example of the usefulness of the properties of the Yukawa interactions . 92 9 Conclusion 95 V Appendix A - Basic notions in group theory 97 10 General 99 10.1 Group . 99 10.2 Order of a group . 99 10.3 Order of an element . 99 10.4 Rearrangement lemma . 99 10.5 Subgroup . 99 10.6 Conjugate elements . 99 10.7 (Conjugacy) class . 100 10.8 Invariant subgroup . 100 10.9 Cosets . 100 10.10Factor group . 100 10.11Direct product group . 100 10.12Classification of groups . 100 10.12.1 Abelian group . 100 10.12.2 Finite group . 100 10.12.3 Isomorphic groups . 100 10.12.4 Simple and semi-simple groups . 100 10.12.5 Lie Group . 100 11 Group Representations 101 11.1 Representation of a group . 101 11.2 Faithful representation . 101 11.3 Equivalent representations . 101 11.4 Invariant subspace . 101 11.5 Irreducible representation . 101 11.6 Unitary representation . 102 x 11.7 Direct sum representation . 102 11.8 Direct product representation . 102 11.9 Schur’s lemma 1 . 102 11.10Schur’s lemma 2 . 102 11.11Characters of a representation . 102 11.12Orthonormality and completeness relations of irreducible characters . 103 11.13Character table . 103 11.14Reduction of a representation . 103 VI Appendix.
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