Selection of Doubly Cabibbo suppressed $D^{*+} \rightarrow D^0 (K^+ \pi^-) \pi^+$ decays and measurement of the $D^0$ lifetime in $D^0 \rightarrow K^- \pi^+$ at the LHCb experiment

The LHC beauty (LHCb) experiment is one of the four main experiments at the Large Hadron Collider, colliding protons since March 2010 with a center-of-mass energy of 7 TeV. This thesis investigates the current status towards a mixing analysis in the $D$ system in LHCb data. First, a selection of Doubly Cabibbo Suppressed (DCS) $D^0 \rightarrow K^+ \pi^-$ decays in the presently largest LHCb data set is presented. The self tagging decay chain $D^{*+} \rightarrow D^0 \pi^+$ is used. A signal of $1966 \pm 28$ decays is measured in $37 \textrm{pb}^{-1}$ of data. The ratio of DCS decays to Cabibbo Favored (CF) $D^0 \rightarrow K^- \pi^+$ decays, using the same $D^*$ decay chain, is measured with $(4.89 \pm 0.07) \times 10^{-3}$. Second, the $D^0$ lifetime is measured in the CF decay mode. In hadronic interactions the proper time distribution of heavy mesons is distorted due to trigger cuts on the final state hadrons' impact parameters. In this thesis an average proper time acceptance function is evaluated in the Monte Carlo (MC) simulation to correct for the distortion. Therefore, the track resolution in MC needed to be adapted to the track resolution measured in data. The $D^0$ proper time resolution is determined in MC to be $(50.89 \pm 11.38) \textrm{fs}$. The fraction of $D^0$ coming from $B$ decays is measured with $(5.16 \pm 0.20) \%$ in data. The $D^0$ lifetime is measured in data with $\tau = (451.8 \pm 3.1 (\textrm{stat.}) \pm 6.5 (\textrm{syst.})) \textrm{fs}$.