Measurements of charmless b-hadron decays at CDF; first evidence for the annihilation $B^0_s \to \pi^+ \pi^-$ decay mode.
In this thesis I report on several results obtained from an accurate analysis of the $Bhh$ data sample: Annihilation decay modes: using 6 fb$^{-1}$ of data collected by the CDF experiment, I obtained the first evidence of the $B_s^0\to \pi^+ \pi^-$ decay mode, with a significance of about 3.7$\sigma$, and I performed the most precise measurement of the Branching Ratio of the $B_d^0\to K^+ K^-$ decay mode at the time, corresponding to an excess of 2$\sigma$ level. These measurements have important consequences as discussed in the previous sections: the comparison between these branching fractions and the theoretical predictions provides unique information for tuning the phenomenological models. Our results prompted intense theoretical work both using pQCD and QCDF approaches to accommodate the discrepancies between predictions and experimental results. CP violation: using the full CDF 10 fb$^{-1}$ of data collected by the CDF experiment, I found evidence of the \textbf{CP violation} of $B_s^0 \to K \pi$ decay mode, with a significance of 3$\sigma$, confirming the result recently obtained at LHCb. I also measured the CP asymmetries of the $B_d^0 \to K \pi$ decay modes, in agreement and of comparable resolution with the b-factories experiments, while the most precise measurement up to date is from LHCb. The precision reached in the Acp($B_s^0 \to K \pi$ ) measurement gives the possibility to perform interesting checks of the Standard Model predictions, explained in the previous sections. I also obtained the world's best measurements of the CP asymmetries in $\Lambda_b^0 \to p\pi$ and $\Lambda_b^0 \to pK$ decay modes. These measurements are unique and represent a step forward in the understanding of the \CP\ violation phenomena in the relatively unexplored $\Lambda_b^0$ sector.