Model Building and Phenomenology in Grand Unified Theories Tom´asE. Gonzalo Velasco University College London Submitted to University College London in fulfillment of the requirements for the award of the degree of Doctor of Philosophy August 2015 2 Declaration I, Tom´asE. Gonzalo Velasco, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. Tom´asE. Gonzalo Velasco August 2015 3 4 Abstract The Standard Model (SM) of particle physics is known to suffer from several flaws, and the upcoming generation of experiments may shed some light onto their solution. Whether there is evidence of new physics or not, theories Beyond the SM (BSM) must be able to accommodate and explain the coming data. The lack of signs of BSM physics so far, calls for a exhaustive exploration beyond the minimal models, in particular Grand Unified theories, for they are able to solve some of the issues of the SM and can make testable predictions. Therefore, we attempt to develop a framework to build Grand Unified models, capable of generating and analysing general non-minimal models. In order to do so, first we create a computational tool to handle the group theoretical component, calculating properties of Lie Groups and their representations. Among them, those of interest to the model building process are the calculation of breaking chains from a group to a subgroup, the decomposition of representations of a group into those of a subgroup and the construction of group invariants. Using some of the capabilities of the group tool, and starting with a set of representations and a breaking chain, we generate all the conceivable models, classifying them to satisfy conditions such as anomaly cancellation and symmetry breaking. We then move on to study the unification of gauge couplings on the models and its consequences on the scale of unification and the scale of supersymmetry breaking, to later constrain them to match phenomenological observables, such as proton decay or current collider searches. We conclude by focusing the analysis on two specific models, a minimal supersymmetric SO(10) model, with some interesting predictions for future colliders, and a flipped SU(5) ⊗ U(1) model, which serves as the triggering mechanism for the end of the inflationary epoch in the early universe. 5 6 Acknowledgements First and foremost, I would like to thank my supervisor, Frank Deppisch, for his support during the very long journey that has been my Ph.D. His expert guidance and sound advice have proven essential for my research and the writing of this thesis. In addition, I would like to thank John Ellis who provided the means and opportunity, and whose vast knowledge helped me greatly along the way. Also, I would like to express my gratitude to all my colleagues and friends at UCL, past and present, for the good times and fun these past years. My sincere gratitude to Luk´aˇsGr´af,Julia Harz and Wei Chih Huang, with whom it was a pleasure to work. Without their help and contribution this thesis would not be finished. I must also thank my family: my parents, Conchita and Tom´as,and my sisters, Conchi and Ester, who have always inspired and supported me, even from the dis- tance. Lastly, I would like to thank Angela,´ for the unwavering support and infinite patience, and for giving me the necessary strength to see this through to the end. 7 8 9 The history of science shows that theories are perishable. With every new truth that is revealed we get a better understanding of Nature and our conceptions and views are modified. - Nikola Tesla 10 Contents 1 Introduction 21 2 Gauge Models in Particle Physics 25 2.1 The Standard Model . 26 2.2 Grand Unified Theories . 32 2.2.1 Georgi-Glashow Model: SU(5) . 34 2.2.2 Flipped SU(5) ⊗ U(1) . 37 2.2.3 Pati-Salam Model . 40 2.2.4 Left-Right Symmetry . 43 2.2.5 SO(10) . 44 2.3 Supersymmetry . 51 2.3.1 Introduction to Supersymmetry . 52 2.3.2 Supersymmetry Breaking . 55 2.3.3 The MSSM . 57 3 Symmetries and Lie Groups 65 3.1 Definition . 66 3.2 Lie algebras . 67 3.3 Representations . 71 3.4 Cartan Classification of Simple Lie Algebras . 75 3.5 Group Theory Tool . 83 3.5.1 Roots and Weights . 85 3.5.2 Subgroups and Breaking Chains . 90 11 Contents 12 3.5.3 Decomposition of Representations . 96 3.5.4 Constructing Invariants . 102 3.6 Implementation and Example Run . 105 3.6.1 C++ Backend . 106 3.6.2 Mathematica Frontend . 110 3.6.3 Sample Session . 112 4 Automated GUT Model Building 117 4.1 Generating Models . 118 4.2 Model Constraints . 121 4.2.1 Chirality . 122 4.2.2 Cancellation of Anomalies . 122 4.2.3 Symmetry Breaking . 125 4.2.4 Standard Model . 126 4.3 Unification of Gauge Couplings . 126 4.3.1 Supersymmetry . 129 4.3.2 Abelian Breaking . 130 4.3.3 Solving the RGEs . 131 4.4 Results for Intermediate Left-Right Symmetry . 132 4.4.1 Proton Decay . 134 4.4.2 Direct and Indirect Detection Constraints . 136 4.4.3 Model Analysis . 138 5 Aspects of GUT Phenomenology 147 5.1 Minimal SUSY SO(10) . 149 5.1.1 The Model . 150 5.1.2 Renormalisation Group Equations . 154 5.1.3 Direct SUSY Searches at the LHC . 157 5.1.4 Phenomenological Analysis . 160 5.2 Flipped GUT Inflation . 170 13 Contents 5.2.1 Inflation . 171 5.2.2 Minimal GUT Inflation . 176 5.2.3 Embedding in SO(10) . 184 6 Conclusions and Outlook 187 A MSSM RGEs 191 A.1 Gauge Couplings . 191 A.2 Yukawa Couplings . 191 A.3 Gaugino Masses . 192 A.4 Trilinear Couplings . 192 A.5 Scalar Masses . 193 A.6 µH and B Terms . 195 A.7 Two-Loop Corrections . 196 Contents 14 List of Figures 1.1 RGE running of the Standard Model gauge couplings . 22 1.2 RGE running of the MSSM gauge couplings . 23 2.1 Breaking patterns of the Pati-Salam model . 41 2.2 Patters of symmetry breaking from SO(10) to the SM group . 46 2.3 One-loop contributions to the Higgs mass . 51 2.4 Searches for SUSY and current limits from the ATLAS collaboration 63 2.5 Searches for SUSY and current limits from the CMS collaboration . 64 3.1 Dynkin diagrams of simple Lie algebras . 81 3.2 Algorithm for obtaining the roots of a simple algebra . 86 3.3 Algorithm for calculating the weights of a representation . 88 3.4 Extended or affine Dynkin diagrams of simple Lie algebras . 90 3.5 Obtaining the SO(5) × SU(2) × U(1) subalgebra of SO(9) . 91 3.6 Algorithms for obtaining the maximal subalgebras of a Lie algebra . 92 3.7 Obtaining the SU(4) ⊗ SU(2) ⊗ SU(2) subalgebra of SO(10) . 92 3.8 Algorithm to obtain the breaking chains of a Lie algebra . 94 3.9 Example of breaking chains from SO(10) to SU(3) ⊗ SU(2) ⊗ U(1) . 95 3.10 Algorithms for calculating the projection matrix . 98 3.11 Algorithm for identifying the representations from the subweights . 100 3.12 Algorithm for calculating the direct product of representations . 103 3.13 Class diagram of the group theory tool . 106 3.14 File system for the group theory tool . 107 15 List of Figures 16 3.15 Properties of the group SU(4) ⊗ SU(2) ⊗ SU(2) . 115 3.16 Properties of some representations of SU(4) ⊗ SU(2) ⊗ SU(2) . 116 4.1 Algorithm for generating models . 121 4.2 Triangle diagram for Adler-Bell-Jackiw anomalies . 122 4.3 Feynman diagram for the main decay modes of protons through di- mension 6 operators . 134 4.4 Feynman diagram for the main decay modes of protons through di- mension 5 operators in a SUSY GUT . 135 4.5 Running of the gauge couplings in a sample scenario of the left-right symmetry model . 139 4.6 Dependence of the scales in a sample scenario of the left-right sym- metry model . 140 4.7 Histograms of models, no constraints . 141 4.8 Histograms of models, with respect to MLR ............... 143 4.9 Histograms of models, with respect to MSUSY ............. 144 4.10 Histograms of models, with respect to MGUT .............. 145 4.11 2D histogram of models in the (MSUSY ;MLR) plane . 146 5.1 Planck exclusion limits on inflationary models . 148 2 5.2 First generation sfermions masses as function of mD .......... 153 5.3 RGE running of 1st generation scalar, gaugino and Higgs doublet masses . 156 5.4 Comparison of exclusion limits on squark masses for the CMSSM and simplified scenarios with the ATLAS limit . 159 2 5.5 Sparticle masses as a function of mD for the scenario with light 3rd generation . 162 2 5.6 Exclusion areas for stau, sbottom and selectron masses in the (mD; m1=2) 2 and (mD;A0) planes for the scenario with light 3rd generation . 163 5.7 Sparticle spectrum for a scenario with light 3rd generation . 164 2 5.8 Sparticle masses as function of mD for the scenario with light 1st generation . 165 17 List of Figures 2 5.9 Exclusion areas for stau, sbottom and selectron masses in the (mD; m1=2) 2 and (mD;A0) planes for the scenario with 1st first generation . 166 5.10 Sparticle spectrum for a scenario with light 1st generation .