Isospin analysis of charmless B-meson decays

We discuss the determination of the CKM angle $\alpha$ using the non-leptonic two-body decays $B\to \pi\pi$, $B\to \rho\rho$ and $B\to \rho\pi$ using the latest data available. We illustrate the methods used in each case and extract the corresponding value of $\alpha$. Combining all these elements, we obtain the following determination of $\alpha_{\rm dir}=( 86.2^{+4.4}_{-4.0}~~ \cup~~ 178.4^{+3.9}_{-5.1})^\circ$. We assess the uncertainties associated to the breakdown of the isospin hypothesis and the choice of statistical framework in detail. We also determine the hadronic amplitudes (tree and penguin) describing the QCD dynamics involved in these decays, briefly comparing our results with theoretical expectations. For each observable of interest in the $B\to \pi\pi$, $B\to \rho\rho$ and $B\to \rho\pi$ systems, we perform an indirect determination based on the constraints from all the other observables available and we discuss the compatibility between indirect and direct determinations. Finally, we review the impact of future improved measurements on the determination of $\alpha$.