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Textures, Model Building, and Orbifold Gauge Anomalies: Research in Three Topics in Physics Beyond the Standard Model

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TEXTURES, MODEL BUILDING, AND ORBIFOLD GAUGE ANOMALIES: RESEARCH IN THREE TOPICS IN PHYSICS BEYOND THE STANDARD MODEL DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Leslie J. Schradin III, B.S., M.S. ***** The Ohio State University 2006 Dissertation Committee: Approved by Professor Stuart Raby, Adviser Professor Richard Kass Adviser Professor Junko Shigemitsu Graduate Program in Professor Terrence Walker Physics ABSTRACT We introduce the Standard Model, list a large sector of the low energy data, and present extensions to the Standard Model including grand unification, supersymme- try, and orbifold extra dimensions. These foundations underly the research presented in this dissertation, which is from three separate projects. Texture models are Ans¨atze for the undiagonalized Yukawa matrices in which some of the matrix elements have been chosen to vanish. Recent precise measurements of sin 2β from the B-factories (BABAR and BELLE) and a better known strange quark mass from lattice QCD make precision tests of predictive texture models possible. We show that in a set of these models, their maximal sin 2β values rule them out at the 3σ level. While at present sin 2β and V /V are equally good for testing N-zero | ub cb| texture models, in the near future the former will surpass the latter in constraining power. We construct a supersymmetric SO(10) D grand unified model with an orbifold × 3 1 extra dimension S /(Z Z′ ). The model uses 11 parameters to fit the 13 independent 2 × 2 low energy observables of the charged fermion Yukawa matrices and predicts the val- ues of two quark mass combinations, mu/mc and mdmsmb, to each be approximately 1σ above their experimental values. The remaining observables are successfully fit at the 5% level. This model is shown to have a gauge anomaly on one of the fixed points and we discuss the alterations in field content necessary to repair it. ii Extra dimensional orbifold theories have gauge anomaly structures which are more complicated than those of Minkowski space. We review previous work done by von Gersdorff and Quiros to derive general expressions for orbifold gauge anomalies. These equations are applied to a supersymmetric 6D orbifold model with E6 gauge symme- try presented by Kobayashi, Raby, and Zhang in order to verify the gauge anomaly cancellations. From this illustration we conclude that the constraining power of orb- ifold gauge anomalies on the field content of the theory is about as great as the usual case in Minkowski space and depends highly on the gauge groups and number of dimensions present. iii For my parents. iv ACKNOWLEDGMENTS There are many people who contributed in many ways to my graduate educa- tion. In the area of research, I’d like to thank my advisor Stuart Raby and fellow collaborator Hyung Do Kim for giving me the opportunity to work with them. My research would not have been possible without their guidance. I am also indebted to the rest of Stuart’s high energy group over the last several years, particularly Radovan Dermisek, Motomichi Harada, and Akin Wingerter for discussions over my research. In the area of my dissertation, I owe thanks to those past graduate students who created the osudissert96 Latex template I used to create this document. I also thank Patrick Randerson and Ashish Saxena for their help with questions on the construction of my dissertation, Stuart Raby for input on content, Ed Smith for his services as a courier, and my dissertation committee for their time and effort. My understanding of physics and mathematics was formed in the classrooms of Ohio State, Yale, Turpin, and Sherwood. Many fine teachers contributed to my education and I would not be where I am today without their instruction. I am grateful for the many friends and acquaintances I have made over the years of graduate school. In particular I am glad for my friendship with Patrick and Susan Blanderson. My parents and family are my foundation. Their constant love, encouragement, and belief in me through all of my years have created the person I am today. v Most importantly, I am thankful to Meredith Howard. Her friendship, support, understanding, and love helped me to survive the black times and to celebrate the occasional successes that made up graduate school. She provided useful advice on most aspects of my written dissertation and defense presentation, including discus- sions over content and emphasis, Latex formatting, and the use of PowerPoint. I would not have made it without her. vi VITA October 2, 1975 ............................Born - Cincinnati, Ohio, USA 1998 ........................................B.S., Physics, Yale University 2003 ........................................M.S., Physics, The Ohio State University 1999-2002 ..................................Graduate Fellow, The Ohio State University 2002-present ................................Graduate Teaching or Research Asso- ciate, The Ohio State University PUBLICATIONS Research Publications H.D. Kim, S. Raby, and L. Schradin, “Quark Mass Textures and sin 2β”, Physical Review D 69, 092002 (2004). H.D. Kim, S. Raby, and L. Schradin, “Quark and Lepton Masses in 5D SO(10)”, The Journal of High Energy Physics 0505, 036 (2005). FIELDS OF STUDY Major Field: Physics Studies in Physics beyond the Standard Model: Professor Stuart Raby vii TABLE OF CONTENTS Page Abstract....................................... ii Dedication...................................... iv Acknowledgments.................................. v Vita ......................................... vii ListofTables.................................... xii ListofFigures ................................... xiv Chapters: 1. Introduction.................................. 1 1.1 Quantum Field Theory and Gauge Symmetry . 1 1.2 TheStandardModel.......................... 1 1.3 ExtensionstotheStandardModel . 3 1.4 OriginalWork ............................. 5 2. StandardModel................................ 7 2.1 TheStandardModelDefined . 7 2.2 Deficiencies of the Standard Model . 14 2.2.1 NeutrinoMasses ........................ 15 2.2.2 HierarchyProblem . 17 2.2.3 DarkMatter .......................... 18 2.2.4 TheUnexplained ........................ 18 2.2.5 OtherProblems......................... 19 viii 3. Data...................................... 20 3.1 ChargedFermionMasses. 20 3.2 CKMelements ............................. 22 3.3 GaugeSector .............................. 24 4. ExtensionstotheStandardModel . 25 4.1 GrandUnifiedTheories(GUTS) . 25 4.1.1 Predictions ........................... 26 4.1.2 Problems ............................ 29 4.1.3 Groups ............................. 30 4.2 Supersymmetry(SUSY) . 35 4.2.1 Motivation ........................... 35 4.2.2 SomeFinerPoints ....................... 36 4.2.3 Minimal Supersymmetric Standard Model . 39 4.3 FamilySymmetry ........................... 41 4.4 ExtraDimensionsandOrbifolds . 42 4.5 Conclusion ............................... 44 5. Texture Models and Sin 2 β ......................... 45 5.1 Introduction .............................. 45 5.2 Setup .................................. 48 5.3 Analysis................................. 51 5.3.1 2 2LightQuarkMatrices . 51 × 5.3.2 3 3QuarkMatrices ..................... 51 × 5.4 Discussion ............................... 57 5.4.1 ComparisonoftheTwoCases. 57 5.4.2 FutureProspects ........................ 58 6. 5D SUSY SO(10) D GUTModel..................... 59 × 3 6.1 Introduction .............................. 59 6.2 Setup .................................. 66 6.2.1 Background .......................... 66 6.2.2 YukawaMatrix ........................ 73 6.3 Analysis ................................ 82 6.3.1 AnalyticFitting ........................ 82 6.3.2 NumericalFitting . .. .. 93 6.4 Justificationin6D ........................... 97 6.5 Pati-SalamBraneGaugeAnomaly . 101 ix 6.6 SummaryandDiscussion . 102 7. OrbifoldAnomalies.............................. 106 7.1 IntroductionandMotivation. 106 7.2 AnomalyBackground . .. .. 108 7.2.1 FujikawaMethod . .. .. 109 7.2.2 GaugeTheories ........................ 113 7.2.3 ExtraDimensions . 117 7.3 OrbifoldAnomalies .......................... 119 7.3.1 Orbifolds ............................ 119 7.3.2 OrbifoldAnomalyEquations . 123 7.4 Applications .............................. 131 7.4.1 Model A1 of Kobayashi, Raby, and Zhang . 131 7.4.2 5DModel ........................... 151 7.5 Discussion ............................... 158 8. Conclusion................................... 163 Appendices: A. LorentzSpinors................................ 167 A.1 General Notation in d Dimensions .................. 167 A.2 d =4Dimensions............................ 169 A.3 d =6Dimensions............................ 172 B. Diagonalization of 2 2Matrices...................... 177 × B.1 GeneralNotation............................ 177 B.2 Approximate Diagonalization for Hierarchical Matrices ....... 178 B.3 Exact Diagonalization for 12 = 21 and11=0........... 179 | | | | C. LoopRephasingMethod ........................... 184 D. D3 FamilySymmetry............................. 193 E. Massless States and Wavefunctions . 196 x F. ALifted4DModel .............................. 198 G. CalculationofSpatialTraces. 209 H. CalculationofGroupTraces . 211 Bibliography .................................... 217 xi LIST OF TABLES Table Page 2.1 TheStandardModelfields.. .. .. 9 2.2 The independent Standard Model parameters. ... 15 4.1 The Minimal Supersymmetric Standard
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