\contentsline {chapter}{{Abstract}}{ii}{chapter*.1}
\contentsline {chapter}{{Contents}}{iii}{chapter*.2}
\contentsline {chapter}{{List of Tables}}{vii}{chapter*.3}
\contentsline {chapter}{{List of Figures}}{viii}{chapter*.4}
\contentsline {chapter}{{Acknowledgements}}{x}{chapter*.5}
\contentsline {part}{I\hspace {1em}{Introduction, Physics Motivation and Apparatus}}{1}{part.1}
\contentsline {chapter}{\numberline {1}{Introductory Topics}}{2}{chapter.1}
\contentsline {section}{\numberline {1.1}{Discovery of the Muon}}{2}{section.1.1}
\contentsline {section}{\numberline {1.2}{Muon Production}}{3}{section.1.2}
\contentsline {section}{\numberline {1.3}{The Standard Model of Particle Physics}}{5}{section.1.3}
\contentsline {subsection}{\numberline {1.3.1}{Elementary Particles}}{5}{subsection.1.3.1}
\contentsline {subsection}{\numberline {1.3.2}{Quantum Chromodynamics}}{6}{subsection.1.3.2}
\contentsline {subsection}{\numberline {1.3.3}{Electroweak Interaction}}{6}{subsection.1.3.3}
\contentsline {section}{\numberline {1.4}{Muon Decay}}{9}{section.1.4}
\contentsline {subsection}{\numberline {1.4.1}{General Four-Fermion Point Interaction }}{9}{subsection.1.4.1}
\contentsline {subsection}{\numberline {1.4.2}{Previous $P_{\mu }\xi $ Measurements}}{10}{subsection.1.4.2}
\contentsline {subsection}{\numberline {1.4.3}{Muon Decay Asymmetry in Angle (MDAA)}}{12}{subsection.1.4.3}
\contentsline {subsubsection}{{MDAA: Integral Asymmetry}}{12}{section*.6}
\contentsline {subsubsection}{{MDAA: Asymmetry}}{13}{section*.7}
\contentsline {section}{\numberline {1.5}{Brief Review of Muon Spin}}{14}{section.1.5}
\contentsline {subsection}{\numberline {1.5.1}{Basics of Spin of Leptons }}{14}{subsection.1.5.1}
\contentsline {subsection}{\numberline {1.5.2}{Quantization of Spin }}{15}{subsection.1.5.2}
\contentsline {subsection}{\numberline {1.5.3}{Motion of Spin in a Uniform External Magnetic Field }}{16}{subsection.1.5.3}
\contentsline {subsection}{\numberline {1.5.4}{Surface Muon Spin}}{16}{subsection.1.5.4}
\contentsline {subsection}{\numberline {1.5.5}{The Bargman-Michel-Telegdi Equation}}{17}{subsection.1.5.5}
\contentsline {section}{\numberline {1.6}{Energy Loss and Muon Depolarization }}{19}{section.1.6}
\contentsline {section}{\numberline {1.7}{Physics Motivation}}{21}{section.1.7}
\contentsline {subsection}{\numberline {1.7.1}{Model Independent Tests }}{21}{subsection.1.7.1}
\contentsline {subsection}{\numberline {1.7.2}{Left-Right Symmetric Model Tests }}{21}{subsection.1.7.2}
\contentsline {chapter}{\numberline {2}{Apparatus}}{25}{chapter.2}
\contentsline {section}{\numberline {2.1}{TRIUMF Cyclotron and Muon Production}}{25}{section.2.1}
\contentsline {section}{\numberline {2.2}{Muon Beam Selection with M13}}{27}{section.2.2}
\contentsline {subsection}{\numberline {2.2.1}{Muon Production Target}}{28}{subsection.2.2.1}
\contentsline {section}{\numberline {2.3}{TWIST Spectrometer}}{30}{section.2.3}
\contentsline {section}{\numberline {2.4}{Solenoid Magnet and Iron Yoke}}{31}{section.2.4}
\contentsline {section}{\numberline {2.5}{Cradle and Chamber Spacing}}{34}{section.2.5}
\contentsline {section}{\numberline {2.6}{Drift Chambers and $\cal TWIST${} Coordinates}}{35}{section.2.6}
\contentsline {section}{\numberline {2.7}{Proportional Chambers}}{38}{section.2.7}
\contentsline {section}{\numberline {2.8}{DC and PC Calibrations}}{38}{section.2.8}
\contentsline {section}{\numberline {2.9}{Gas System}}{39}{section.2.9}
\contentsline {section}{\numberline {2.10}{Muon Stopping Target}}{40}{section.2.10}
\contentsline {section}{\numberline {2.11}{Beam Package and Muon Ranging}}{41}{section.2.11}
\contentsline {section}{\numberline {2.12}{Time Expansion Chamber and Beam Characterization}}{43}{section.2.12}
\contentsline {subsection}{\numberline {2.12.1}{TEC Basic Layout}}{44}{subsection.2.12.1}
\contentsline {subsection}{\numberline {2.12.2}{TEC Calibration and Performance}}{45}{subsection.2.12.2}
\contentsline {subsubsection}{{TEC Calibrations}}{45}{section*.8}
\contentsline {paragraph}{{STR Correction.}}{45}{section*.9}
\contentsline {paragraph}{{TEC Efficiency Correction.}}{46}{section*.10}
\contentsline {subsection}{\numberline {2.12.3}{Uncertainties in TEC Measurements}}{46}{subsection.2.12.3}
\contentsline {paragraph}{{$t_0$ calibrations.}}{46}{section*.11}
\contentsline {paragraph}{{STR.}}{48}{section*.12}
\contentsline {paragraph}{{Analysis algorithm.}}{48}{section*.13}
\contentsline {paragraph}{{TEC resolution.}}{48}{section*.14}
\contentsline {paragraph}{{Multiple scattering in the TEC.}}{51}{section*.15}
\contentsline {paragraph}{{Alignment of the collimator.}}{51}{section*.16}
\contentsline {paragraph}{{Alignment of TEC to the drift chambers.}}{51}{section*.17}
\contentsline {paragraph}{{TEC measurements during solenoid field ramping}}{51}{section*.18}
\contentsline {paragraph}{{Overall Estimate of TEC Uncertainty}}{53}{section*.19}
\contentsline {subsection}{\numberline {2.12.4}{Muon Beam as Seen by the TEC}}{53}{subsection.2.12.4}
\contentsline {section}{\numberline {2.13}{Electronics and Trigger Logic}}{55}{section.2.13}
\contentsline {part}{II\hspace {1em}{Analysis, Data, Simulation, and Systematic Effects}}{56}{part.2}
\contentsline {chapter}{\numberline {3}{Michel Spectrum Analysis Strategy}}{57}{chapter.3}
\contentsline {section}{\numberline {3.1}{Analysis Flow}}{57}{section.3.1}
\contentsline {section}{\numberline {3.2}{Unpacking}}{58}{section.3.2}
\contentsline {section}{\numberline {3.3}{Pattern Recognition and Event Classification}}{59}{section.3.3}
\contentsline {subsection}{\numberline {3.3.1}{Calculating TDC Hit Statistics in Bins of Time}}{59}{subsection.3.3.1}
\contentsline {subsection}{\numberline {3.3.2}{Set Window Times}}{60}{subsection.3.3.2}
\contentsline {subsection}{\numberline {3.3.3}{Look for Outlier Noise Hits at Large Radius}}{61}{subsection.3.3.3}
\contentsline {subsection}{\numberline {3.3.4}{Calculate TDC Hit Statistics in Window Times}}{61}{subsection.3.3.4}
\contentsline {subsection}{\numberline {3.3.5}{Assign Hits to the Window Structure}}{61}{subsection.3.3.5}
\contentsline {subsubsection}{{Evaluate Window Types One at a Time}}{62}{section*.20}
\contentsline {subsubsection}{{Review Window Types Looking at Whole Event}}{64}{section*.21}
\contentsline {subsubsection}{{Assign Hits to Window}}{64}{section*.22}
\contentsline {subsection}{\numberline {3.3.6}{Decide on Overall Event Type}}{64}{subsection.3.3.6}
\contentsline {section}{\numberline {3.4}{Clustering}}{68}{section.3.4}
\contentsline {section}{\numberline {3.5}{Track Fitting}}{69}{section.3.5}
\contentsline {subsection}{\numberline {3.5.1}{Initial Estimate}}{69}{subsection.3.5.1}
\contentsline {subsection}{\numberline {3.5.2}{Helix Fitting}}{70}{subsection.3.5.2}
\contentsline {section}{\numberline {3.6}{Energy Calibration}}{72}{section.3.6}
\contentsline {subsection}{\numberline {3.6.1}{Endpoint Energy Calibration Method}}{72}{subsection.3.6.1}
\contentsline {subsection}{\numberline {3.6.2}{Endpoint Energy Calibration Results}}{74}{subsection.3.6.2}
\contentsline {subsection}{\numberline {3.6.3}{Momentum Dependent Energy Calibration}}{75}{subsection.3.6.3}
\contentsline {section}{\numberline {3.7}{Track Selection and Cuts}}{76}{section.3.7}
\contentsline {subsection}{\numberline {3.7.1}{Tree Processing Code Description}}{76}{subsection.3.7.1}
\contentsline {subsection}{\numberline {3.7.2}{Fiducial Cuts}}{81}{subsection.3.7.2}
\contentsline {chapter}{\numberline {4}{Data}}{83}{chapter.4}
\contentsline {section}{\numberline {4.1}{Data Sets}}{83}{section.4.1}
\contentsline {section}{\numberline {4.2}{2002, 2003 and 2005 Data}}{85}{section.4.2}
\contentsline {section}{\numberline {4.3}{ Results of 2005 Data Solenoid Field Change }}{88}{section.4.3}
\contentsline {section}{\numberline {4.4}{Data Quality}}{89}{section.4.4}
\contentsline {chapter}{\numberline {5}{Simulation}}{92}{chapter.5}
\contentsline {section}{\numberline {5.1}{Muon Decay Spectrum and Radiative Corrections}}{92}{section.5.1}
\contentsline {section}{\numberline {5.2}{Geometry}}{93}{section.5.2}
\contentsline {section}{\numberline {5.3}{Chamber Response}}{93}{section.5.3}
\contentsline {section}{\numberline {5.4}{Beam Input}}{94}{section.5.4}
\contentsline {section}{\numberline {5.5}{Spin Tracking}}{95}{section.5.5}
\contentsline {section}{\numberline {5.6}{Muon Depolarization in Material}}{96}{section.5.6}
\contentsline {subsection}{\numberline {5.6.1}{Gaussian Depolarization}}{96}{subsection.5.6.1}
\contentsline {subsection}{\numberline {5.6.2}{Exponential Depolarization}}{96}{subsection.5.6.2}
\contentsline {section}{\numberline {5.7}{Validation}}{98}{section.5.7}
\contentsline {chapter}{\numberline {6}{Systematic Uncertainty Descriptions and Estimates}}{101}{chapter.6}
\contentsline {section}{\numberline {6.1}{Systematic Uncertainty Due to Muon Beam Characteristics and Polarization}}{103}{section.6.1}
\contentsline {subsection}{\numberline {6.1.1}{Depolarization in the Muon Production Target }}{103}{subsection.6.1.1}
\contentsline {subsection}{\numberline {6.1.2}{Depolarization in the Beamline Window Valve}}{104}{subsection.6.1.2}
\contentsline {subsection}{\numberline {6.1.3}{Fringe Field Depolarization (FFD)}}{104}{subsection.6.1.3}
\contentsline {subsubsection}{{FFD: Beam Position and Angle Uncertainty }}{105}{section*.23}
\contentsline {subsubsection}{{FFD: Different Field Maps}}{109}{section*.24}
\contentsline {subsubsection}{{FFD: Deconvolution of TEC Multiple Scattering }}{109}{section*.25}
\contentsline {subsubsection}{{FFD: Muon Beam Size}}{110}{section*.26}
\contentsline {subsection}{\numberline {6.1.4}{Muon Depolarization at the Stopping Location (MDSL)}}{113}{subsection.6.1.4}
\contentsline {subsubsection}{{MDSL: Mechanisms for Muon Depolarization in Materials}}{113}{section*.27}
\contentsline {subsubsection}{{MDSL: Materials in the Vicinity of the Muon Stopping Target}}{114}{section*.28}
\contentsline {subsubsection}{{MDSL: Estimate from Extrapolation Error}}{116}{section*.29}
\contentsline {subsection}{\numberline {6.1.5}{Background Muon Contamination }}{119}{subsection.6.1.5}
\contentsline {subsection}{\numberline {6.1.6}{Beam Intensity and Coincident Accidental Particles}}{119}{subsection.6.1.6}
\contentsline {subsection}{\numberline {6.1.7}{Proton Beam Stability }}{120}{subsection.6.1.7}
\contentsline {section}{\numberline {6.2}{Systematic Uncertainty Due to Chamber Response}}{121}{section.6.2}
\contentsline {subsection}{\numberline {6.2.1}{Wire to Wire Time Variations}}{121}{subsection.6.2.1}
\contentsline {subsection}{\numberline {6.2.2}{Chamber Foil Bulge}}{121}{subsection.6.2.2}
\contentsline {subsection}{\numberline {6.2.3}{Chamber Cell Asymmetry}}{122}{subsection.6.2.3}
\contentsline {subsection}{\numberline {6.2.4}{Density Variations}}{122}{subsection.6.2.4}
\contentsline {subsection}{\numberline {6.2.5}{Muon Dead Zone}}{122}{subsection.6.2.5}
\contentsline {subsection}{\numberline {6.2.6}{Upstream versus Downstream Efficiency}}{123}{subsection.6.2.6}
\contentsline {section}{\numberline {6.3}{Systematic Uncertainty Due to Momentum Calibration}}{125}{section.6.3}
\contentsline {subsection}{\numberline {6.3.1}{Endpoint Fits}}{125}{subsection.6.3.1}
\contentsline {subsection}{\numberline {6.3.2}{Magnetic Field Map Accuracy}}{125}{subsection.6.3.2}
\contentsline {section}{\numberline {6.4}{Systematic Uncertainty Due to Positron Interactions}}{127}{section.6.4}
\contentsline {subsection}{\numberline {6.4.1}{Energy Loss}}{127}{subsection.6.4.1}
\contentsline {subsection}{\numberline {6.4.2}{Hard and Intermediate Interactions }}{127}{subsection.6.4.2}
\contentsline {subsection}{\numberline {6.4.3}{Multiple Scattering}}{130}{subsection.6.4.3}
\contentsline {subsection}{\numberline {6.4.4}{Material Outside the Detector }}{131}{subsection.6.4.4}
\contentsline {subsection}{\numberline {6.4.5}{Stopping Target Thickness}}{131}{subsection.6.4.5}
\contentsline {section}{\numberline {6.5}{Systematic Uncertainty Due to Spectrometer Alignment}}{133}{section.6.5}
\contentsline {subsection}{\numberline {6.5.1}{Longitudinal Alignment }}{133}{subsection.6.5.1}
\contentsline {subsection}{\numberline {6.5.2}{Rotational Alignment}}{133}{subsection.6.5.2}
\contentsline {subsection}{\numberline {6.5.3}{Magnetic Field Alignment}}{133}{subsection.6.5.3}
\contentsline {subsection}{\numberline {6.5.4}{Translational Alignment}}{133}{subsection.6.5.4}
\contentsline {section}{\numberline {6.6}{Set-Dependent Systematic Uncertainties}}{134}{section.6.6}
\contentsline {part}{III\hspace {1em}{Results and Conclusion}}{135}{part.3}
\contentsline {chapter}{\numberline {7}{Results}}{136}{chapter.7}
\contentsline {section}{\numberline {7.1}{Blind Fit Results}}{136}{section.7.1}
\contentsline {subsection}{\numberline {7.1.1}{Blind Fit for P$_{\mu }\xi ${}}}{136}{subsection.7.1.1}
\contentsline {subsection}{\numberline {7.1.2}{Blind Fit Results for $\rho $ and $\delta $}}{141}{subsection.7.1.2}
\contentsline {section}{\numberline {7.2}{Consistency Check and Unblinding}}{143}{section.7.2}
\contentsline {subsection}{\numberline {7.2.1}{ Consistency Check Results }}{143}{subsection.7.2.1}
\contentsline {subsection}{\numberline {7.2.2}{ Uncertainty due to Eta }}{143}{subsection.7.2.2}
\contentsline {section}{\numberline {7.3}{Final Results}}{144}{section.7.3}
\contentsline {subsection}{\numberline {7.3.1}{Left-Right Symmetric Model Limits}}{146}{subsection.7.3.1}
\contentsline {chapter}{\numberline {8}{Future Improvements and Conclusion}}{147}{chapter.8}
\contentsline {section}{\numberline {8.1}{Future Improvements}}{147}{section.8.1}
\contentsline {section}{\numberline {8.2}{Conclusion}}{147}{section.8.2}
\contentsline {chapter}{{Bibliography}}{148}{chapter*.30}
\contentsline {chapter}{\numberline {A}{Personal Contributions to $\cal TWIST${}}}{153}{appendix.A}
\contentsline {section}{\numberline {A.1}{Contributions to Design and Building}}{153}{section.A.1}
\contentsline {section}{\numberline {A.2}{Contributions to Operation of $\cal TWIST${}}}{153}{section.A.2}
\contentsline {section}{\numberline {A.3}{Contributions to Analysis of $\cal TWIST${} Data}}{153}{section.A.3}