$CP$ violation and branching fraction measurements of three-body charmless $B^{\pm}$ decays at LHCb

Three independent analysis are reported based on a data sample corresponding to an integrated luminosity of 3 fb$^{-1}$ of $pp$ collisions recorded with LHCb detector. Firstly, the relative branching fractions of $B^+ \to h^+ h'^+ h'^-$ decays, where $h^{(')}$ is a pion or kaon, are measured. The results obtained improve significantly on previous measurements of these quantities, and are important for the interpretation of Dalitz plot analyses of three-body charmless hadronic decays of $B^+$ mesons. Furthermore, the amplitude analysis of $B^{\pm} \to \pi^{\pm} K^+ K^-$ and $B^{\pm} \to \pi^{\pm} \pi^+ \pi^-$ are also reported. Regarding the $B^{\pm} \to \pi^{\pm} K^+ K^-$ decay, the rescattering amplitude produces a negative $\mathit{CP}$ symmetry of $(-66 \pm 4 \pm 2)\%$, which is the largest $\mathit{CP}$ violation effect observed from a single amplitude. Whereas in the $B^{\pm} \to \pi^{\pm} \pi^+ \pi^-$ decay, a large $\mathit{CP}$ asymmetry is observed in the decay amplitude involving the tensor $f_2(1270)$ resonance, as well as in the $\pi^+ \pi^-$ \textit{S}-wave at low invariant mass. The presence of $\mathit{CP}$ violation related to interference between the $\pi^+ \pi^-$ \textit{S}-wave and the \textit{P}-wave $B^+ \to \rho(770)^0 \pi^{\pm}$ amplitude is also established.