Tracking optimisation and the measurement of $K^+ \to \pi^+ \mu^+ \mu^-$ at NA62

The NA62 experiment at CERN is designed to perform a precision measurement of the ultra-rare $K^+ \to \pi^+ \nu \bar\nu$ branching fraction. The experiment relies on a kinematic background rejection using the missing mass-squared signatures of the contributing background components. As the signal is of the $\cal O$($10^{-10})$ smaller than that of the most common kaon decays, the tracking performance of the detector must be exceptional in order to achieve a large signal to background ratio of $S/B \approx 10$. In this thesis, two tools to aid with the tracking procedure have been developed. Firstly, a field map which describes the fringe field of the spectrometer magnet has been created, tested and implemented in the NA62 Monte-Carlo software. Additionally, an analysis procedure to improve the precision and accuracy of track momentum in data was developed. Such tools are essential, not only for the $K^+ \to \pi^+ \nu \bar\nu$ decay, but also for all decays containing tracks. In particular, they have been used to aid the measurement of the branching fraction and the form factor for the Standard Model decay $K^+ \to \pi^+ \mu^+ \mu^-$ which is described here. The measurement has been performed using the data collected by the NA62 experiment in 2016 and 2017. The event selection, trigger efficiency determination and the analysis techniques used for the measurement are presented. The analyses of the two data sets are provided separately to demonstrate the progressing sensitivity of the detector. The final results are compatible with literature, and the 2017 measurement is currently the world's most precise for this channel. This demonstrates the future potential of NA62 for when larger data sets become available in the next few years.