Search for the rare decay $\Lambda_{b}\rightarrow p \pi^{-} \mu^{+}\mu^{-}$

This thesis reports the branching fraction measurement of the rare Cabibbo-suppressed decay $\Lambda_{b}\rightarrow p\pi^{-}\mu^{+}\mu^{-}$. The decay is observed for the first time with a $5.5\sigma$ deviation from the background-only hypothesis. This is the first observation of a $b\rightarrow d$ quark transition in the baryon sector. The dataset used for the measurement corresponds to 3 $\mathrm{fb}^{-1}$ of $pp$ collisions collected at the LHCb experiment at CERN. The branching fraction is measured using $\Lambda_{b}\rightarrow J/\psi(\rightarrow\mu^{+}\mu^{-}) p \pi^{-}$ as a normalisation channel and is measured as \begin{equation*} \mathcal{B}(\Lambda_{b}\rightarrow p\pi^{-}\mu^{+}\mu^{-}) = (6.9 \pm 1.9 \pm 1.1^{+1.3}_{-1.0})\times 10^{-8}, \end{equation*} where the first error is the statistical uncertainty, the second is the systematic uncertainty and the third is the uncertainty on $\mathcal{B}(\Lambda_{b}\rightarrow J/\psi(\rightarrow\mu^{+}\mu^{-}) p \pi^{-})$. The measurement of $ \mathcal{B}(\Lambda_{b}\rightarrow p\pi^{-}\mu^{+}\mu^{-})$ can be combined with the branching fraction measurement for $\Lambda_{b}\rightarrow pK^{-}\mu^{+}\mu^{-}$ to give constraints on the ratio of CKM matrix elements $|\frac{V_{td}}{V_{ts}}|$. Such a determination of $|\frac{V_{td}}{V_{ts}}|$ requires a th\ eory prediction for the ratio of the relevant form factors. This thesis also reports the ratio of tracking efficiencies, $\epsilon_{\mathrm{rel}}$, between data and simulation for $K_{s}\rightarrow\pi^{+}\pi^{-}$ decays occurring within the LHCb detector acceptance. As \KS particles are long-lived, their associ\ ated tracking efficiencies are less precisely determined compared to those of shorter-lived particles. The average value of $\epsilon_{\mathrm{rel}}$ for $K_{s}\rightarrow\pi^{+}\pi^{-}$ decays, where the $K_{s}$ has a flight distance of $\sim 1m$, is found to be \begin{equation*} \epsilon_{\mathrm{rel}} = 0.70\pm0.02. \end{equation*} To perform this calibration measurement a novel technique was developed which has the potential to be used in measuring the value of $\epsilon_{\mathrm{rel}}$ for other decays involving long-lived particles.