Measurement of the forward-backward asymmetry ($A_\textit{FB}$) of $Z$ boson decays in electron-positron pairs with the ATLAS detector in proton-proton collisions at $\sqrt{s}$ = 7 TeV at the LHC

In this thesis the measurement of the effective weak mixing angle $\sin^2{\theta_W^{\mathrm{eff}}}$ in proton-proton collisions is described. The results are extracted from the forward-backward asymmetry ($A_\textit{FB}$) in electron-positron final states at the ATLAS experiment at the LHC. The $A_\textit{FB}$ is defined upon the distribution of the polar angle between the incoming quark and outgoing lepton. The signal process used in this study is the reaction $pp\to Z/\gamma^*+X\to e^+e^-+X$ taking a total integrated luminosity of 4.8 fb$^{-1}$ of data into account. The data was recorded at a proton-proton center-of-mass energy of $\sqrt{s}=7$ TeV. The weak mixing angle is a central parameter of the electroweak theory of the Standard Model (SM) and relates the neutral current interactions of electromagnetism and weak force. The higher order corrections on $\sin^2{\theta_W^{\mathrm{eff}}}$ are related to other SM parameters like the mass of the Higgs boson. Because of the symmetric initial state constellation of colliding protons, there is no favoured forward or backward direction in the experimental setup. The reference axis used in the definition of the polar angle is therefore chosen with respect to the longitudinal boost of the electron-positron final state. This leads to events with low absolute rapidity have a higher chance of being assigned to the opposite direction of the reference axis. This effect called dilution is reduced when events at higher rapidities are used. It can be studied including electrons and positrons in the forward regions of the ATLAS calorimeters. Electrons and positrons are further referred to as electrons. To include the electrons from the forward region, the energy calibration for the forward calorimeters had to be redone. This calibration is performed by inter-calibrating the forward electron energy scale using pairs of a central and a forward electron and the previously derived central electron energy calibration. The uncertainty is shown to be dominated by the systematic variations. The extraction of $\sin^2{\theta_W^{\mathrm{eff}}}$ is performed using $\chi^2$ tests, comparing the measured distribution of $A_\textit{FB}$ in data to a set of template distributions with varied values of $\sin^2{\theta_W^{\mathrm{eff}}}$. The templates are built in a forward folding technique using modified generator level samples and the official fully simulated signal sample with full detector simulation and particle reconstruction and identification. The analysis is performed in two different channels: pairs of central electrons or one central and one forward electron. The results of the two channels are in good agreement and are the first measurements of $\sin^2{\theta_W^{\mathrm{eff}}}$ at the $Z$ resonance using electron final states at proton-proton collisions at $\sqrt{s}=7$ TeV. The precision of the measurement is already systematically limited mostly by the uncertainties resulting from the knowledge of the parton distribution functions (PDF) and the systematic uncertainties of the energy calibration. The extracted results of $\sin^2{\theta_W^{\mathrm{eff}}}$ are combined and yield a value of $\sin^2{\theta_W^{\mathrm{eff}}}_{\rm comb} = 0.2288 \pm 0.0004 ({\rm stat.}) \pm 0.0009 ({\rm syst.}) = 0.2288 \pm 0.0010 ({\rm tot.})$. The measurements are compared to the results of previous measurements at the $Z$ boson resonance. The deviation with respect to the combined result provided by the LEP and SLC experiments is up to 2.7 standard deviations.