Studies of charmless three-body $b$-hadron decays at LHCb

Studies of charmless three-body decays of either $B^{0}_{(s)}$ mesons or beauty baryons with a $K^{0}$ meson in the final state are presented in this thesis. The analyses are performed using the 2011 and 2012 LHCb dataset, corresponding to an integrated luminosity of $1.0\,$fb$^{-1}$ and $2.0\,$fb$^{-1}$ recorded at a centre-of-mass energy of $7\,$TeV and $8\,$TeV, respectively. The decays $B^{0}_{s}\rightarrow K^{0} K^{\pm} \pi^{\mp}$ and $B^{0}_{s} \rightarrow K^{0} \pi^{+}\pi^{-}$ are observed for the first time, and the decay mode $B^{0} \rightarrow K^{0} K^{\pm}\pi^{\mp}$, previously observed by the BaBar experiment, is confirmed. Moreover, the $\Lambda^{0}_{b} \rightarrow \bar{K}^{0} p \pi^{-}$ channel is also observed for the first time and its phase-space integrated "CP" asymmetry measured, which shows no significant deviation from zero. No significant signals are seen for $\Lambda^{0}_{b} \rightarrow K^{0} p K^{-}$ decays, $\Xi^{0}_{b}$ decays to both the $K^{0}_{\rm S}p\pi^{-}$ and $K^{0}_{\rm S}pK^{-}$ final states, and the $\Lambda^{0}_{b} \rightarrow D^{-}_{s}(K^{0}_{\rm S}K^{-})p$ decay, and upper limits on their branching fractions are reported. Finally, the first untagged decay-time-integrated amplitude analysis of $B^{0}_{s}\rightarrow K^{0}_{\rm S}K^{\pm}\pi^{\mp}$ decays has been performed. All branching fraction results obtained from this analysis are either the most precise to date or are first measurements.