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Running Neutrino Masses and Flavor Symmetries Michael Andreas Schmidt Max-Planck-Institut für Kernphysik Saupfercheckweg 1 D-69117 Heidelberg E-Mail: [email protected] 50 Jahre 1958 2008 Max-Planck-Institut für Kernphysik Technische Universität München Physik Department Institut für Theoretische Physik T30d Running Neutrino Masses and Flavor Symmetries Dipl.-Phys. Univ. Michael Andreas Schmidt Vollständiger Abdruck der von der Fakultät für Physik der Technischen Universität München zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzender: Univ.-Prof. Dr. Reiner Krücken Prüfer der Dissertation: 1. Prof. Dr. Manfred Lindner, Ruprecht-Karls-Universität Heidelberg 2. Univ.-Prof. Dr. Andrzej J. Buras Die Dissertation wurde am 10. Juni 2008 bei der Technischen Universität München eingereicht und durch die Fakultät für Physik am 30. Juni 2008 angenommen. Running Neutrino Masses and Flavor Symmetries Abstract The flavor structure of neutral fermion masses is completely different from charged fermion masses. The cascade seesaw framework allows to implement a cancellation mechanism, which leads to a weak hierarchy of neutrinos despite the large hierarchy in the charged fermion masses. We present one realization by the gauge group E6 and two in the framework of SO(10) based on discrete flavor symmetries T7 and Σ(81). Higher-dimensional operators as well as the flavon potential are discussed. Furthermore, since renormalization group (RG) effects can become very important in neutrino physics, especially in view of upcoming precision experiments, we have investigated several models. The Lµ − Lτ flavor symmetry in the standard seesaw scenario leads to quasi-degenerate neutrinos and therefore to large RG corrections. The quantum corrections to quark-lepton-complementarity (QLC) relations are extensively discussed. In the minimal supersymmetric standard model (MSSM), the effect is almost always positive. In the standard model (SM), there are sizable RG corrections due to the threshold effects which are either positive for ∆' ≈ 0◦ or negative for ∆' ≈ 180◦. They have been studied in the leading log (LL) approximation and mainly lead to a rescaling of right-handed (RH) neutrino masses. The results are generalized beyond LL and the conditions for the applicability are derived. Finally, the results are applied to the cascade seesaw mechanism and the cancellation mechanism. The RG equations of the mixing parameters in the triplet seesaw are derived in terms of basis-independent quantities. The main results are the independence of Majorana phases and the proportionality to the mass squared difference in the strongly hierarchical case which differs from the standard seesaw mechanism. Kurzfassung Die Familienstruktur der neutralen Fermionen unterscheidet sich völlig von den geladenen Fermion- massen. Das kaskadierte Seesaw Szenario erlaubt die Konstruktion eines Auslöschungsmechanismus, der zu einer schwachen Hierarchie der Neutrinos führt trotz der starken Hierarchie der geladenen Fermionen. Wir präsentieren eine Realisierung durch die Eichgruppe E6 und zwei im Kontext von SO(10), die auf einer diskreten Familiensymmetrie T7 bzw. Σ(81) basieren. Höher-dimensionale Operatoren und das Flavonpotential werden diskutiert. Da Renormierungsgruppen(RG)-Effekte in der Neutrinophysik sehr wichtig werden können, insbesondere im Blick auf die kommenden Präzis- sionsexperimente, wurden mehrere Modelle untersucht. Die Lµ −Lτ Symmetrie im Standard Seesaw Modell führt zu quasi-degenerierten Neutrinos und somit zu großen RG Effekten. Die Quantenko- rrekturen zu den Quark-Lepton Komplementarität Relationen werden ausführlich diskutiert. Im Minimalen Supersymmetrischen Modell (MSSM) sind die Korrekturen fast immer positiv. Im Stan- dard Modell (SM), gibt es größere RG Korrekturen aufgrund von Schwelleneffekten, die positiv für ∆' ≈ 0◦ bzw. negativ für ∆' ≈ 180◦ sind. Sie werden in der führenden Logarithmus Näherung be- sprochen und führen hauptsächlich zu einer Reskalierung der rechtshändigen (RH) Neutrinomassen. Die Ergebnisse werden über die LL Näherung hinaus verallgemeinert und Bedingungen der An- wendbarkeit werden hergeleitet. Schließlich werden die Ergebnisse beim kaskadierten Seesaw Mech- anismus und dem Auslöschungsmechanismus angewandt. Die RG Gleichungen der Mischungspa- rameter im Triplett Seesaw Mechanismus werden hergeleitet und basisunabhängig ausgedrückt. Die Hauptergebnisse sind die Unabhängigkeit von den Majoranaphasen und die Proportionalität der Massenquadratdifferenzen im stark hierarchischen Fall im Gegensatz zum Standard Seesaw Mech- anismus. Contents 1 Introduction 1 2 Renormalization Group 5 2.1 Basic Picture – Wilson Renormalization Group . .5 2.2 Callan-Symanzik Equation . .7 2.3 Effective Field Theories . .8 2.4 Decoupling . 10 3 Aspects of Model Building 11 3.1 Neutrino Masses . 11 3.1.1 Experimental Data . 11 3.1.2 Neutrino Mass Models . 13 3.2 Unified Theories . 17 3.2.1 SO(10) ........................................ 19 3.2.2 E6 .......................................... 22 3.3 Flavor Symmetries . 23 3.3.1 Continuous Symmetries . 23 3.3.2 Discrete Symmetries . 27 3.4 Quark Lepton Complementarity . 29 4 Cancellation Mechanism 33 4.1 Description of the Mechanism . 33 4.2 Singular MSS ........................................ 34 4.3 Realization of DS Matrix Structure . 35 4.4 Realization of Cancellation Mechanism with GUT Symmetry . 36 4.5 Realization of Cancellation Mechanism with Flavor Symmetry . 38 4.5.1 T7 Realization . 39 4.5.2 Σ(81) Realization . 45 5 Threshold Corrections 51 5.1 Thresholds in the Standard (Type I) Seesaw Model . 51 5.1.1 Effects of Vertex Corrections . 55 5.1.2 Generalizations . 56 5.2 Thresholds in the Cascade Seesaw . 58 5.3 RG Stability of the Cancellation Mechanism . 59 5.3.1 Singular MSS .................................... 60 5.3.2 Quasi-Degenerate Neutrino Spectrum . 61 5.3.3 Perturbations of MSS ............................... 63 5 6 CONTENTS 5.3.4 T7 Realization . 64 5.3.5 Σ(81) Realization . 65 6 RG Effects in Neutrino Mass Models 67 6.1 General Structure of RG Equations in Seesaw Models . 67 6.2 Lµ − Lτ Flavor Symmetry . 69 6.3 Quark Lepton Complementarity . 72 6.3.1 RG Effects: General Considerations . 72 6.3.2 RG Effects: MSSM Case . 73 6.3.3 RG Effects: SM Case . 77 6.3.4 Renormalization of 1-3 Mixing . 78 6.3.5 Level Crossing Points . 79 6.3.6 Evolution above the GUT scale . 81 6.4 Triplet (Type II) Seesaw Model . 82 6.4.1 Running of Masses . 82 6.4.2 Running of Mixing Angles . 83 6.4.3 Running of Phases . 85 6.4.4 RG Evolution in Type I+II Model . 86 7 Summary & Conclusions 87 A Conventions 91 A.1 Mixing Matrices . 91 A.2 Standard Parameterization . 92 B Group Theory 93 B.1 Lie Groups . 93 B.1.1 SO(10) . 94 B.1.2 E6 .......................................... 95 B.2 Discrete Groups . 98 B.2.1 T7 .......................................... 98 B.2.2 Σ(81) ........................................ 101 B.3 Anomalies . 105 B.3.1 Anomalies of Gauge Theories . 105 B.3.2 Discrete Anomalies . 105 C Renormalization Group 107 C.1 General RG Equations for Mixing Parameters . 107 C.2 RG Factors in the Standard Seesaw Model . 111 C.2.1 SM . 111 C.2.2 MSSM . 112 Acknowledgments 113 Bibliography 114 Chapter 1 Introduction The quantization of charge and the unification of the gauge couplings of all three forces in the 16 minimal supersymmetric standard model (MSSM) at ΛGUT = 2·10 GeV strongly suggest a further unification of all forces into a (supersymmetric (SUSY)) grand unified (GU) gauge group [1, 2]. Especially, models which unify all SM particles in one irreducible representation like SO(10) [3, 4] and E6 [5–9] are appealing. The flavor sector also shows regularities: (i) All charged fermion masses show a strong normal hierarchy. (ii) The mixing angles in the quark sector are small and in the lepton sector, there is maximal 2-3 mixing and possibly vanishing 1-3 mixing. (iii) The neutrino mass matrix is compatible with the exchange of the second and third row/column which suggests a µ − τ exchange symmetry. Furthermore, there are some peculiar relations: (i) The light quark masses are related to the Cabibbo angle r md tan #12 ≈ (1.1) ms 1 which is known as Gatto-Sartori-Tonin (GST) relation [11]. (ii) The quark mixing angles #ij and the lepton mixing angles θij, which parameterize the Cabibbo-Kobayashi-Maskawa (CKM) and Maki-Nakagawa-Sakata (MNS) mixing matrix, respectively, add up to maximal mixing π π θ + # ≈ ; θ + # ≈ : (1.2) 12 12 4 23 23 4 They are known as quark lepton complementarity (QLC) relations [12–14] because of the com- plementarity of θij and #ij to maximal mixing. They suggest a further unification of quarks and leptons. Therefore the QLC relations are presumably due to a symmetry close to the unification scale. (iii) Moreover, to a very high precision (10−5), the charged lepton masses fulfill 2 p p p m + m + m = m + m + m 2 ; (1.3) e µ τ 3 e µ τ which was found by Koide [15]. It indicates that the first generation is as important as the third one, which is in contradiction to the approach to generate the masses of the third generation at tree level and the light generations by non-renormalizable operators. Here, all flavors should be considered as equally important. All these structures and relations of the experimental data indicate, that the masses and mixing angles are not accidental but determined by a flavor symmetry. There already exist several attempts to explain theses patterns by continuous (e.g. Abelian [16–24] and non-Abelian [25–29]) or discrete 1It can be explained by a symmetric mass matrix with vanishing 1-1 element [10], e.g. within SO(10). 1 2 CHAPTER 1. INTRODUCTION (e.g. A4 [30]) symmetries. However, the study of flavor symmetries has revealed, that they have to be broken, explicitly or spontaneously, above the electroweak scale since the low-energy data does not allow for an exact symmetry. In view of these hints towards a GU theory (GUT) as well as a flavor symmetry, it is tempting to combine a GU gauge group and a flavor symmetry.
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    Proceedings of the 2017 CERN–Latin-American School of High-Energy Physics, San Juan del Rio, Mexico 8-21 March 2017, edited by M. Mulders and G. Zanderighi, CYR: School Proceedings, Vol. 4/2018, CERN-2018-007-SP (CERN, Geneva, 2018) Beyond the Standard Model M. Mondragon Instituto de Fisica, Universidad Nacional Autonoma de Mexico UNAM, CD MX 01000, Mexico Abstract I present a brief outline of some of the different paths for extending the Stan- dard Model. These include supersymmetry, Grand Unified Theories, extra dimensions, and multi-Higgs models. The aim is to give graduate students an overview of some of the theoretical motivations to go into a specific extension, and a few examples of how the experimental searches go hand in hand with these efforts. Supersymmetry, Grand Unified Theories, Extra dimensions, Multi-Higgs models 1 Introduction The Standard Model is an extremely successful description of the elementary particles and its interac- tions around the Fermi scale, but it leaves a number of unanswered questions that lead naturally to the conclusion that it is the low energy limit of a more fundamental theory. Along with these unanswered question comes a large number of free parameters whose value can only be determined experimentally. Among the unanswered questions in the Standard Model are: – Are there more than three generations of particles? – Why are the fundamental particles masses so different? – How is the Higgs mass stablilized, i.e. what solves the hierarchy problem? – Is there more than one Higgs? – What is the nature of the neutrinos, are they Dirac or Majorana? – What is dark matter? – Why is there more matter than anti-matter? – Is there more CP violation? – Why only left-handed particles feel the electroweak interaction? – Is it possible to unify quantum mechanics with gravity? The collection of models and theories that make the research that extends the Standard Model to answer some of these, and many more, questions is known as physics Beyond the Standard Model (BSM).