Charmless hadronic three-body decays of neutral $B$ mesons with a $K^0_s$ in the final state in the LHCb experiment: branching fractions and an amplitude analysis: Désintégrations hadroniques à trois corps sans charme de mésons $B$ avec un $K^0_s$ dans l'état final dans l'expérience LHCb : mesure de rapports d'embranchement et une analyse en amplitude

This dissertation presents several studies of the decays of both $B_d$ and $B_s$ mesons to charmless three-body final states including a $K_{\rm S}^0$ meson. They use the data recorded by the LHCb experiment during Run I of LHC, corresponding to an integrated luminosity of $\int\mathcal{L}=3\rm fb^{-1}$. A first analysis consists of the measurement of the branching fractions of $B_{d,s}\rightarrow K_{\rm S}^0h^{+}h^{'-}$ decays, where $h^{(')}$ designates a kaon or a pion. Preceding LHCb measurements of branching fractions for all decay channels, relative to that of $B_d\rightarrow K_{\rm S}^0\pi^{+}\pi^{-}$, are updated. Furthermore, the primary goal of this analysis is to search for the, as yet, unobserved decay $B_s\rightarrow K_{\rm S}^0K^+K^-$. The relative branching fractions are measured to be: \begin{align} \frac{\mathcal{B}\left(B_s\rightarrow K_{\rm S}^0\pi^{+}\pi^{-}\right)}{\mathcal{B}\left(B_d\rightarrow K_{\rm S}^0\pi^{+}\pi^{-}\right)} &= 0.26 \pm 0.04\mathrm{(stat.)} \pm 0.02\mathrm{(syst.)} \pm 0.01\mathrm{(f_s/f_d}\mathrm{)},\\ \frac{\mathcal{B}\left(B_d\rightarrow K_{\rm S}^0K^{\pm}\pi^{\mp} \right)}{\mathcal{B}\left(B_d\rightarrow K_{\rm S}^0\pi^{+}\pi^{-}\right)} &= 0.17 \pm 0.02\mathrm{(stat.)} \pm 0.00\mathrm{(syst.)},\\ \frac{\mathcal{B}\left(B_s\rightarrow K_{\rm S}^0K^{\pm}\pi^{\mp} \right)}{\mathcal{B}\left(B_d\rightarrow K_{\rm S}^0\pi^{+}\pi^{-}\right)} &= 1.84 \pm 0.07\mathrm{(stat.)} \pm 0.02\mathrm{(syst.)} \pm 0.04\mathrm{(f_s/f_d}\mathrm{)},\\ \frac{\mathcal{B}\left(B_d\rightarrow K_{\rm S}^0K^{+}K^{-} \right)}{\mathcal{B}\left(B_d\rightarrow K_{\rm S}^0\pi^{+}\pi^{-}\right)} &= 0.59 \pm 0.02\mathrm{(stat.)} \pm 0.01\mathrm{(syst.)},\\ \end{align} A first observation of $B_s\rightarrow K_{\rm S}^0K^{+}K^{-}$ is reported with a global significance of $3.7 \sigma$. A flavour-untagged, time-independent Dalitz-plot analysis of $B_d\rightarrow K_{\rm S}^0K^{+}K^{-}$ is presented, using the isobar approach. The quasi-two-body branching fractions of $B_d\rightarrow K_{\rm S}^0\phi^{0}$, $B_d\rightarrow K_{\rm S}^0f_2^{'}(1525)$, $B_d\rightarrow K_{\rm S}^0 f_0(1710)$, and $B_d\rightarrow K_{\rm S}^0\chi_c^{0}$ are measured. They are compatible with previous measurements from BaBar, except for $B_d\rightarrow K_{\rm S}^0 f_0(1710)$.