Measurement of $sin2\beta$ from $B^{0}_{d} \rightarrow J/\Psi K^{0}_{S}$ with the LHCb detector

The LHCb experiment will study CP violation and rare B meson decays at the LHC accelerator, in the CERN laboratories of Geneva. The CP violation in the $b$ quark mesons was observed for the first time measuring the $\sin2\beta$ parameter. In the first year of data taking at LHCb the $\sin2\beta$ measurement will be performed in the channel $B_d ^0 \rightarrow J/\psi K_S ^0$ (called golden channel). The comparison with results obtained in previous experiments will show possible systematic errors. Once the systematic errors will be understood, the possible discrepancies with the expected results will be the sign of a new physics beyond the Standard Model. $\sin2\beta$ is extracted from the asymmetry in the decay $B_d ^0 \rightarrow J/\psi(\mu^+\mu^-) K_S ^0 (\pi^+\pi^-)$ and its CP conjugated $\overline{B}_d ^0 \rightarrow J/\psi(\mu^-\mu^+) K_S ^0(\pi^-\pi^+)$. To distinguish between $B_d ^0$ and $\overline{B}_d$ flavour tagging algorithms are used. These introduce an additional parameter: the wrong tag fraction ($\omega_{tag}$). As a consequence the measured parameter is $(1-2\omega_{tag})\sin2\beta$. $\omega_{tag}$ is obtained from the asymmetry measure of the control channel $B_d ^0 \rightarrow J/\psi(\mu^+\mu^-) K^{0*}(K^+\pi^-)$. In this thesis the various aspects of the $\sin2\beta$ measurement at LHCb are presented. The data from the LHCb simulation were analysed to tune the selection of the golden channel and the control channel. The applied selection does not modify the decay time distribution of the B meson. If the distribution of this variable is modified, an acceptance function must be introduced. This function can't be extracted from experimental data. The selection is also unified, which means that it is as similar as possible for the two channels. At the end of the selection the values of $\omega_{tag}$ for the two channels were compared, and it was verified that the results of the control channel can be applied to the golden channel. On the selected events a model was studied to describe the distr ibutions of mass and decay time of the B meson. This is a set of probability density functions ($pdf$) which describe the mass and decay time distributions for the signal events and the different background categories. The decay time $pdf$ depends on the $\sin2\beta$ parameter. The mass is used to discriminate between signal and background events. Combining these distributions a total $pdf$ with 16 parameters is obtained, and it will be used to fit real data. To study the LHCb sensitivity to $\sin2\beta$ a toy Monte Carlo was performed. A number of events equal to the one selected in one year of data taking at LHCb ($10^{7}$ s) were generated following the total $pdf$ distribution. From these data and the total $pdf$ a likelihood function was made. From the minimization of this function an estimation of $\sin2\beta$ and its error was obtained . Moreover it was calculated the sensitivity varying the number of background events. My study uses an accurate description of the background and a selection that does not modify the decay time distribution.