Reconstruction of the Channel $B^0_d \rightarrow D^+ \pi^-$ and Background Classification at LHCb (revised)

The channel $B^0_d \rightarrow D^+ \pi^-$ allows for a theoretically clean extraction of the CKM angle $\gamma$, while its high branching ratio makes it attractive for the measurement of the $B^0_d$ lifetime. A reconstruction strategy is developed for this channel and optimized for three different yield and purity scenarios. A genetic algorithm is developed to allow a quick and robust optimization of the selection cuts. A strategy for classifying the background events passing the offline selection is developed, implemented and deployed to study the types of background expected for this and other channels at LHCb. When optimized for maximal $\sqrt{\frac{S}{S+B}}$, the channel is expected to have a yield of $1.340M \pm 80k$ fully triggered events in 2 fb$^{-1}$, with a signal to background ratio of $4.6 \pm 1.4$. When optimized for purity, the channel is expected to have a yield of $724k \pm 47k$ fully triggered events in 2 fb$^{-1}$, with a signal to background ratio of $20 \pm 6$. This revision corrects an error in the branching ratio for the channel $B^0_d \rightarrow D^+ \pi^-$ used in the previous version of this note.