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High Precision Spin Tune Determination at the Cooler Synchrotron in Jülich

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High Precision Spin Tune Determination at the Cooler Synchrotron in Jülich High Precision Spin Tune Determination at the Cooler Synchrotron in Jülich Der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University vorgelegte Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften vorgelegt von Master of Science Physiker Dennis Eversmann aus Tönisvorst. High Precision Spin Tune Determination at the Cooler Synchrotron in Jülich Additional sources of the violation of the CP symmetry are required in order to explain the predominance of the matter in our universe. This mechanism is directly interlinked to physics beyond the Standard Model and it is one of the most relevant questions in modern particle physics. Prominent candidates to solve this problem are the permanent electric dipole mo- ments (EDMs) of elementary particles like electrons, neutrons or protons. Experiments with neutral particles already started in the middle of the past century and the current results do not differ significantly from a zero EDM. For charged particles like protons and deuterons, an experimental setup at an electrostatic storage ring is proposed. Corresponding feasibility studies are performed by the JEDI (Jülich Electric Dipole moments Investigation) collaboration at the Cooler Synchrotron (COSY) in order to estimate essential requirements and systematic limitations. In the scope of this thesis two important quantities are discussed, the spin tune and the spin coherence time. The first one is defined as the number of spin rotations during one turn of the particle bunch in the storage ring. It is shown, that the quantity can be determined in real time with a high statistical precision, which allows the investigation of systematic effects with unprecedented accuracy. The spin coherence time denotes a statistical limitation. It is a measure of how long the spins of the particles are in phase, while they precess in the horizontal 29 plane. In order to reach a statistical limit of an EDM measurement in the order of 10− e cm, the quantity is needed to be larger than 1 000 s. This work demonstrates how this requirement is met by reducing the spin tune spread using sextupole magnets. The main focus of this thesis is on establishing a theoretical foundation of the related data analysis. Therefore, a naive Bayes approach is used in order to determine the relevant observ- ables, like the phase or the amplitude of the polarization. In addition, a statistical model is developed, which describes the time depending decoherence of the particle spins and the drift of spin tune. This leads to a better understanding of the electromagnetic components in the storage ring and the orbit of the particle beam. ii Contents 1 Introduction 1 1.1 Baryon Asymmetry . .3 1.2 Electric Dipole Moments . .3 1.3 CP violation and EDMs . .4 1.4 EDM Experiments . .5 1.4.1 Charged Particle Experiment . .6 2 Fundamental Statistic and Accelerator Concepts 9 2.1 Statistics . .9 2.1.1 Central Limit Theorem . .9 2.1.2 Bayes Theorem . .9 2.1.3 Estimator . 10 2.1.4 Fisher Information and Cramer-Rao Bound . 12 2.1.5 Maximum Likelihood Estimator . 13 2.1.6 Directional Statistics . 14 2.2 Beam and Spin Dynamics in Storage Rings . 16 2.2.1 Coordinate System and Phase Space . 16 2.2.2 Equations of Motion . 18 2.2.3 Transverse and Longitudinal Motion . 19 2.2.4 Beam Emittance and Betatron Tune . 20 2.2.5 Dispersion and Chromaticity . 22 2.2.6 Spin and Polarization Formalism . 24 2.2.7 Thomas-BMT Equation . 27 2.2.8 Vertical and Horizontal Polarization . 28 3 Experimental Setup and Data Acquisition 31 3.1 The COSY Storage Ring . 32 3.2 Data Acquisition . 35 3.2.1 Time Stamping System . 35 3.2.2 Frequency of the RF Cavity . 35 3.3 Event Rate . 37 3.4 Vertical Asymmetry . 38 3.4.1 Two Polarization States . 39 iii 4 Data Analysis 41 4.1 Mapping Method . 42 4.1.1 Spin Phase Advance js ............................ 42 4.1.2 Up-Down Asymmetries . 44 4.2 Discrete Turn Fourier Transform . 47 4.2.1 Fourier Transform . 47 4.2.2 Discrete Fourier Transform . 47 4.3 Parameter Estimation . 51 4.3.1 Standard Error Estimation . 53 4.3.2 Conclusion . 55 4.4 Probability Density Function . 56 4.4.1 Marginal Probability Density Function of Spin Tune and Phase . 57 4.4.2 Joint Bivariate Probability Density Function . 59 4.4.3 Marginal Probability Density Function of Turn Number . 60 4.5 Conclusion . 62 5 Amplitude Determination 65 5.1 Rice Distribution . 66 5.1.1 Moments of Rice Distribution . 68 5.2 Maximum Likelihood Estimator . 70 5.3 Feldmann-Cousin Algorithm . 71 5.4 Comparison of Estimators . 75 5.5 Efficiency of Estimators . 76 5.6 Amplitude Spectrum . 78 5.6.1 Main Lobe . 80 5.6.2 Maximum Likelihood Fit . 80 5.7 Conclusion . 81 6 Phase Determination 83 6.1 Wrapped Probability Density Function . 84 6.1.1 Maximum Likelihood Estimator . 86 6.1.2 Circular Moment Estimator . 86 6.1.3 Confidence Interval of Estimated Phase . 88 6.1.4 Conclusion . 90 6.1.5 Confidence Interval of True Phase . 91 6.2 Phase Spectrum . 93 6.2.1 Maximum Likelihood Fit . 95 6.3 Combination of Individual Detectors . 98 6.3.1 Weighted Average . 98 6.3.2 Discrete Fourier Transform of two Sinusoidal Functions . 99 6.3.3 Asymmetry (Mapping Method) . 100 6.4 Conclusion . 100 6.5 Spin Tune Determination . 101 6.5.1 Global Maximum Likelihood Fit . 101 iv 7 Results 107 7.1 Systematics of the Phase Estimation . 108 7.1.1 Assumed Spin Tune . 109 7.1.2 Number of Bins of the Asymmetry Distribution . 111 7.1.3 Pull Distribution . 112 7.1.4 Conclusion . 115 7.2 General Experimental Considerations . 116 7.2.1 Resonant Spin Flip . 116 7.2.2 Beam Revolution Period . 117 7.2.3 RF Cavity Period . 119 7.2.4 Event Arrival Time . 121 7.3 Extraction Methods . 124 7.3.1 Vertical and Horizontal Bump Extraction . 124 7.3.2 White Noise Extraction . 127 7.3.3 Conclusion . 128 7.4 Spin Tune Investigations (Beam time June/May 2015) . 129 7.4.1 Variation within a Cycle . 129 7.4.2 Long Term Stability . 137 7.4.3 Long Cycle . 141 8 Conclusion 143 A Appendix 145 A.1 List of Runs . 145 A.2 Integrals for Fisher Information for Spin Precession . 146 A.3 Integral of the bivariat Probability Density Function . 149 A.4 Derivation of the Rician Distribution . 149 A.5 Probability Density Distribution of the Sum of Squared Random Variables . 150 A.6 Integral of the Circular Moment Cosinus Term . 151 A.7 Probability Density Function of the True Phase . 153 Bibliography 155 List of Figures 161 List of Tables 167 v CHAPTER 1 Introduction This thesis is written at the Chair of the III. Physikalisches Institut B of the RWTH Aachen university. Moreover, it is established in the section FAME1 of the Jülich Aachen Research Alliance2. The objective of this cooperation is concerned to a better understanding of the matter-antimatter asymmetry in the universe by performing basic physical research in the field of nuclear and particle physics. Solving the puzzle of the observed asymmetry is exceedingly relevant since it holds the key to our very existence. The focus of this thesis is the investigation of permanent electric dipole moments3 of charged elementary particles in the context of a storage ring experiment. The existence of those is pos- tulated in the Standard Model4.
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