Measuring the decay $B^0 \rightarrow \pi^- D_s^+$ with sensitivity to the $b \rightarrow u$ quark transition

In this thesis, the measurement of $\mathcal{B}(B^0 \rightarrow \pi^- D_s^+)$ at LHCb is discussed. $B^0 \rightarrow \pi^- D_s^+$ is a non-leptonic tree decay, which only occurs via the $b \rightarrow u$ quark transition. As a result, it probes the CKM matrix element $|V_{ub}|$. As $B^0 \rightarrow \pi^- D_s^+$ is purely non-leptonic, non-factorisable contributions $|a_{NF}|$ from the strong interaction have to be taken into account. They are estimated using $B^0 \rightarrow D \bar{D}$ decays while taking non-tree (penguin, exchange and penguin annihilation) contributions into account. It is found that $\mathcal{B}(B^0 \rightarrow \pi^- D_s^+) = (26.7\pm 2.0(stat.)\pm 1.5(syst.)\pm1.6(\mathcal{B}))\times 10^{-6}$, which results in $|V_{ub}||a_{NF}| = (3.40\pm 0.14(\mathcal{B})\pm 0.29(ext.))10^{-3}$, where the first uncertainty comes from the measurement of $\mathcal{B}(B^0 \rightarrow \pi^- D_s^+)$ and the second uncertainty from external inputs.