Calibration of the LHCb VELO Detector and Study of the Decay Mode $D^0 \to K^-\mu^+ \nu_{\mu}$

The LHCb experiment, based at the Large Hadron Collider at CERN, is primarily designed to make precision measurements of the decays of heavy flavour hadrons, such as $B$ and $D$ mesons. This thesis is composed of two parts: the first consists of two studies of LHCb's vertex locator (VELO) and the second describes the development of methods for recording the decay $D^0 \to K^- \mu^+ \nu_{\mu}$. The first VELO study involves calibration and monitoring of the gain (i.e. the detector response to input charge from particles). We propose a robust method to measure the gain response of each silicon sensor using calibration bits output by the sensors, and a method to recalibrate the gain simple enough to be followed by non-expert VELO users. This is followed by an investigation into the prospects of using the VELO to perform particle identification using the characteristic energy deposition of each particle species ($\mathrm{d}E/\mathrm{d}x$). Finally, studies into the development of a trigger and a so-called 'stripping line' for recording $D^0 \to K^- \mu^+ \nu_{\mu}$ decays is presented. The relatively high cross-section for charm decays in LHCb means this decay (with a branching fraction of 3 %) occurs frequently, and the challenge of a trigger is to reduce this to a rate acceptable to write to disk. Finally, based on a sample of data from July and August 2011, the measured $q^2$ distribution for this decay is compared to the simple single-pole theoretical model, and the pole mass is measured to be $m_{\textrm{pole}}=2.35^{+0.81}_{-0.35}$ GeV/$c^2$.