A measurement of the $W$ boson cross-section using the LHCb detector

LHCb is a single arm forward spectrometer specially designed to study $b$-physics. It covers the pseudo-rapidity, $\eta$, in the range of $1.9 < \eta < 4.9$. During the year of 2010, the LHCb experiment collected approximately 37 pb$^{-1}$ of data at $\sqrt{s}=7$ TeV. In this thesis, the total and differential cross-sections as a function of the muon pseudo-rapidity for the $W\rightarrow\mu\nu_{\mu}$ process are measured with this 2010 dataset. A study of this process provides an important test of the Standard Model as well as a reduction of the uncertainty on the Parton Distribution Functions. Additionally it provides complementary measurements to those performed by ATLAS and CMS. A $W\rightarrow\mu\nu_{\mu}$ event selection scheme is informed by the $W\rightarrow\mu\nu_{\mu}$ simulation sample. This scheme is applied to the data sample and yields 26891 candidates for the $W\rightarrow\mu\nu_{\mu}$ process. Of these candidates, 15030 are for the $W^{+}\rightarrow\mu^{+}\nu_{\mu}$ process and 11861 are for the $W^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}$ process. A fit is performed to determine the purity of muons from the $W$ bosons. This purity is found to be about 79$\%$. The final cross-section for the $W\rightarrow\mu\nu_{\mu}$ process is measured in a fiducial phase-space in which the muon transverse momentum is greater than 20 GeV/c and the pseudo-rapidity of the muon is in the range $2.0 < \eta < 4.5$. The results of the total cross-sections in the fiducial phase-space for the $W^{+}\rightarrow\mu^{+}\nu_{\mu}$ and $W^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}$ processes are \begin{align} \sigma_{W^{+}\rightarrow\mu^{+}\nu_{\mu}}=890.1\pm 9.2 \pm 26.4 \pm 30.9 \; \rm{pb}\\ \sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}}=687.0\pm 8.1\pm 19.6\pm 23.7 \; \rm{pb} \end{align} where the first uncertainty is statistical, the second uncertainty is systematic and the third uncertainty is due to the luminosity determination. The ratio between the total cross-sections for the $W^{+}\rightarrow\mu^{+}\nu_{\mu}$ and $W^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}$ processes is \begin{align} \frac{\sigma_{W^{+}\rightarrow\mu^{+}\nu_{\mu}}}{\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}}}=1.292\pm 0.020\pm 0.002 \end{align} where the first uncertainty is statistical and the second uncertainty is systematic. The uncertainty due to the luminosity determination is completely cancelled. The measurements of the $W$ cross-sections and their ratio are consistent with theoretical predictions at next-to-next-leading-order (NNLO) with the MSTW08 and JR09 PDF sets. The cross-section ratio prediction with the ABKM09 PDF set is overestimated. The precision of the total $W^{+}$ ($W^{-}$) cross-section measurement is about 4.7$\%$ (4.6$\%$), which is larger than the percentage uncertainty on the $W^{+}$ ($W^{-}$) prediction due to the MSTW08, ABKM09 and JR09 PDF errors: 1.8$\%$, 1.7$\%$ and 2.9$\%$ (2.6$\%$, 1.6$\%$ and 3.2$\%$). However, as the uncertainties due to the luminosity determination on the $W^{\pm}$ cross-section measurements cancel out and systematic uncertainties are correlated, the dominant uncertainty on the cross-section ratio measurement is statistical. The precision of the total cross-section ratio measurement is 1.6$\%$, which is smaller than the percentage uncertainty on the ratio prediction due to the MSTW08 and JR09 PDF errors: 3.3$\%$ and 3.1$\%$. As a result, the measurement on the $W$ cross-section ratio can reduce uncertainties on the predictions due to the MSTW08 and JR09 PDF errors.