A measurement of the Z cross-section at LHCb

The LHCb experiment, one of the four main experiments at the Large Hadron Collider, recorded 37.66 pb$^{-1}$ of 7 TeV proton collision data in 2010. In this thesis, the total cross section and the differential cross section as a function of boson rapidity of the Z $\to \mu^+ \mu^-$ process are measured using 16.5 pb$^{-1}$ of this data. The precise testing of the Standard Model and reduction of uncertainty on Parton Distribution Functions constitute the primary motivations of these measurements. A scheme to select Z $\to \mu^+ \mu^-$ decays has been developed and was justified using simulation. The application of this scheme to the data yielded 830 candidates for the signal process. The events were used to estimate tracking, muon identification and trigger efficiencies with an overall efficiency of 0.71 $\pm$ 0.04. Three main background sources have been identified: the semi-leptonic decays of beauty and charm quarks, the Z $\to \tau^+ \tau^-$ process where both tau leptons decay to muons and the mis-identification of high momentum hadrons as muons. These backgrounds were quantified using mostly data-driven methods and the total background contamination was found to be 1.4 $\pm$ 1.2 events. The result for the total cross section (in the kinematic region specified) is \begin{align} \sigma. Br(\text{Z} \to \mu^+\mu^-)(2 < \eta_{\mu^+},\eta_{\mu^-} < 4.5, P^{\mu}_{T} > 20 \text{ GeV, } 81 \text{ GeV } < M_{\mu\mu} < 101 \text{ GeV}) \notag \\ = 74 \pm 2 \text{(stat.)} \pm 4 \text{(sys.)} \pm 7 \text{(lumi.) pb} \notag \end{align} where $\eta_{\mu^{ -(+)}}$ is the pseudorapidity of the muon (anti-muon) and $\cal {M}_{\mu\mu}$ is the invariant mass of the muon pair. This result and the measured differential cross section are in agreement with NLO theoretical predictions. The precision of this measurement is not yet sufficient to reduce uncertainties on Parton Distribution Funtionsm, PDFs. Reductions of PDF uncertainty will become possible with a larger dataset with a more precisely determined luminosity. Studies to aid the calibration of the Vertex Locator (VELO), a silicon microstrip sub-detector, are also presented. These studies are concerned with in-situ identification of faulty detector channels and investigation of the macroscopic occupancy behaviour of the detector. Methods to identify faulty channels are described and the numbers of faulty channels are measured. The occupancy of the VELO in 7 TeV proton collision data was studied and was found to be in broad agreement with expectation.