The Electric Dipole Moment Challenge

Chapter 1. Symmetry, Physical Laws, and Electric Dipole Moments 7 1.1. Introduction 7 1.2. Force field symmetries 8 1.2.1. Vectors 8 1.2.2. Vectors and pseudovectors 9 1.2.3. Modern analog of the Amp´ere experiment 11 1.3. Why Measure EDMs, Which, and How? 12

Chapter 2. Some Essential Experiments 15 2.1. Neutron EDM Measurements 15 2.2. Penning Traps and Penning-Like Traps 20 2.3. Electron EDM Measurement Using Polar Molecule Enhancement 21

Bibliography 23

Chapter 3. Magnetic Precessions 25 3.1. Cyclotron Rotation, Gyromagnetic Ratio, and Larmor Precession 25 3.1.1. Classical Picture 25 3.1.2. Magnetic Moment Parameterization 26 3.2. Storage Ring EDM Measurement 29 3.3. Spurious Magnetic Precessions 30 3.3.1. Reducing the Influence of Environmental Magnetic Fields 30 3.3.2. Low Frequency Magnetic Fields 30 3.3.3. Neutron EDM Magnetic Precessions 31 3.3.4. Magnetic Noise Comparison of Neutron and Proton EDM Experiments 33

Bibliography 35

Chapter 4. Just Enough Accelerator Physics 37 4.1. Preview 37 4.2. The uniform field ring 37 4.3. Horizontal stability 38 4.4. Vertical stability 40 4.5. Simultaneous horizontal and vertical stability 41 4.6. Dispersion 42 4.7. Momentum compaction 43 4.8. Chromaticity 43 4.9. Transfer matrices 44

3

4 CONTENTS

4.10. Transfer matrices for simple elements 45 4.10.1. Drift space 45 4.10.2. Thin lens 45 4.11. Transfer matrix parameterization 46 4.12. Strong focusing 47 4.13. General Transverse Motion 48 4.13.1. Magnetic Deflections 48 4.13.2. Element Strengths and Deviations 50 4.13.3. Nonlinear Magnetic Field Example 52

Chapter 5. All-Electric Particle Dynamics 55 5.1. Background 55 5.2. Introduction 56 5.3. Particle tracking paradigms 56 5.4. Relativistic kinematics in central force electric field. 59 5.4.1. Solution of the equation of motion. 60 5.4.2. Rescaling of the MP-vector and updating the horizontal slope. 67 5.4.3. Pseudoharmonic description of the motion. 68

Bibliography 71

Chapter 6. The All-Electric Brookhaven, Electron Storage Ring 73 6.1. Introduction 73 6.2. Storage Rings for Frozen Spin Electrons or Protons. 74 6.2.1. Simulation of Storage Ring Commissioning. 75 6.3. The AGS Electron Analogue Ring 76 6.3.1. Reconstruction from Historical BNL documents. 77 6.4. Current day simulation of 1955 machine studies tune plane scan 83

Bibliography 87

Chapter 7. A Self-Magnetometer Storage Ring 89 7.1. Abstract 89 7.2. Introduction 90 7.3. Orbit Equations for the Storage Ring Bottle 91 7.3.1. Distributed Octupole Field 94 7.3.2. Guiding Center Approximation 94 7.3.3. Fast-Slow Approximation 95 7.3.4. Octupole Restrained Motion 96 7.4. Self-Magnetometer Precision 99 7.4.1. Compensation Procedures 99 7.4.2. Calculated Self-Magnetometer Precision 101

Bibliography 107

Chapter 8. Frequency Domain EDM Experiment Design 109 8.1. Introduction 109

CONTENTS 5

8.1.1. Review of fundamentals 109 8.1.2. Physics Justification and Current Status 110 8.2. Proposed Method 111 8.2.1. Definition of “Nominal” EDM 115 8.2.2. The Frequency Domain “Advantage” 117 8.2.3. Tentative Parameters 118 8.3. Error Analysis Strategy 119 8.3.1. Storage Ring as “Charged Particle Trap” 119 8.3.2. Categorization of Error Sources 120 8.3.3. Noise Limited Precision 121 8.4. Spin Precession 123 8.4.1. Field Transformations 123 8.4.2. MDM-Induced Precession in Electric Field 123 8.4.3. EDM-Induced Precession in Electric Field 124 8.5. Conquering ∆Br Field Errors 125 8.5.1. Qualitative Discussion 125 8.6. Roll-Reversal Accuracy 127 8.6.1. Wien Filter Design 127 8.6.2. Roll-Reversal Symmetry 129 8.6.3. Determination of Roll Reversal Accuracy 130 8.6.4. Determination of Roll Reversal Accuracy 130 8.7. Other Calculations 131 8.7.1. Geometric Phase Errors 131 8.8. Recapitulation and Conclusions 131

Bibliography 135

Chapter 9. The Bargmann-Michel-Telegdi Equation 137 9.1. Relativistic Mechanics 138 9.2. Angular momentum 3-vector s 141 9.3. The Momentum-Weighted Spin 4-Vector W 143 9.4. Lorentz Transformation of 4-Spin Components 143 9.5. The Bargmann, Michel, and Telegdi (BMT) Equation 144 9.6. Special Cases of Spin Precession 148 9.6.1. Pure Thomas Precession 148 9.6.2. Frozen Spin Operation of a Magnetic Storage Ring 150

Bibliography 151

Chapter 10. Relativistic Stern-Gerlach Deflection 153 10.1. Introduction 153 10.2. Brief Historical Perspective 155 10.3. Lorentz Force Law 157 10.4. Relativistic Stern-Gerlach Deflection 158 10.5. Deflection Examples 162 10.5.1. Particle Deflection in Electrostatic Separator 162

6 CONTENTS

10.5.2. S-G Deflection in Magnetic Quadrupole 163 10.5.3. S-G Deflection in Electric Quadrupole 165 10.5.4. Spin-Orbit Coupling 165 10.5.5. Spin-Orbit Central Force 166 10.5.6. Spin-Orbit Energy Shift 167 10.6. Practical Observation of S-G Deflection 169 10.7. Stern-Gerlach Deflection of a Relativistic Particle 171 10.8. S-G Specific Beam Preparation 173 10.9. Signal Levels and Background Rejection 175 10.10. Recapitulation and Acknowledgements 178