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The Fermilab Muon $g-2$ experiment is currently preparing for its fourthdata-taking period (Run-4). The experiment-wide effort on the analysis of Run-1data is nearing completion, with the announcement of the first result expectedin the coming months. The final goal of the experiment is to determine the muonmagnetic anomaly to a precision of 140 ppb. This level of precision willprovide indirect evidence of new physics, if the central value agrees with thepreviously-measured value of the magnetic anomaly. Essential in reducing thesystematic uncertainty, through measurements of the muon beam profile, are thein-vacuum straw tracking detectors. A crucial prerequisite in obtainingaccurate distributions of the beam profile is the internal alignment of thetracking detectors, which is described in this thesis. As a result of thisposition calibration, the tracking efficiency has increased by 3%, while thetrack quality increased by 4%. This thesis also discusses an additionalmeasurement that will be made using the tracking detectors: a search for anelectric dipole moment (EDM) of the muon, through the direct detection of anoscillation in the average vertical angle of the electron from the muon decay.An observation of a muon EDM would be evidence of new physics and would providea new source of CP violation in the charged lepton sector. Essential inmeasuring the EDM are accurate and precise estimations of potential non-zeroradial and longitudinal magnetic fields, which were estimated using the Run-1data. In addition, a preliminary analysis using the Run-1 data was undertakento estimate the available precision for the magnetic anomaly measurement usingthe tracking detectors.

Alignment of the straw tracking detectors for the Fermilab Muon $g-2$  experiment and systematic studies for a muon electric dipole moment  measurement