A measurement of $Z$ production using tau final states with the LHCb detector

The LHCb experiment is one of the four main experiments located on the LHC ring. It collected 1 fb$^{-1}$ of data in 2011 at a centre-of-mass energy of 7 TeV. In this thesis, the cross-section for $Z\to\tau\tau$ production is measured using the full 2011 dataset. A study of this process provides complementary measurements to LHCb's existing results in the $Z\to\mu\mu$ and $Z\to ee$ decay modes and tests lepton universality. Additionally it can be used to search for evidence of new physics processes producing high-mass tau pairs. This measurement is performed using four different di-tau final states, split into five analysis channels based on kinematic requirements. A selection scheme is developed for each and the background contributions and efficiences for reconstructing and selecting signal events are determined using primarily data-driven methods. The results for the analysis channels are then combined, with correlations between the channels taken into account. The final measurement, $\sigma_{Z\to\tau\tau}$, represents the cross-section for the production of a tau pair through an intermediate $Z$ boson where the mass of the $Z$ is between 60 and 120 GeV, the transverse momenta of the tau leptons exceed 20 GeV, and the pseudorapidities of the tau leptons are between 2 and 4.5. The result is \begin{equation} \sigma_{Z\to\tau\tau} = 71.4 \pm 3.5 \pm 2.8 \pm 2.5\;\mathrm{pb} \end{equation} where the first uncertainty is statistical, the second is systematic and the third is due to the luminosity determination. The measurement is in agreement with previous results in the other leptonic decay modes of the $Z$ at LHCb, and with theoretical predictions. The VELO is a high precision tracking detector located around the interaction region. One of the key features of tau decays is their decay length and this can be exploited to distinguish them from prompt backgrounds. A precise measurement of the decay length requires a high hit resolution in the VELO. In this thesis studies of the resolution of the VELO are performed. The single hit resolution of the VELO sensors is studied as a function of strip pitch, and a correction to improve the calculation of the impact positions of tracks on VELO sensors is presented. The best single hit resolution, at a strip pitch of 40 $\mu$m and a projected angle of between 10 and 14$^\circ$ is determined to be 3 $\mu$m, while improvements in resolution of up to 30\% for $r$-clusters at low projected angle are achieved by applying the correction to the calculation of the impact positions. The impact parameter resolution is also studied as a function of transverse momentum and found to be approximately 15 $\mu$m for the $x$ and $y$ components of the impact parameter for tracks with transverse momentum greater than 6 GeV.