CP-violation in B → π+π- and the unitarity triangle

We analyze the extraction of weak phases from CP-violation in B → π+π- decays. We propose to determine the unitarity triangle \((\bar{\rho},\bar{\eta})\) by combining the information on mixing-induced CP-violation in B → π+π-, S, with the precision observable sin2β obtained from the CP-asymmetry in B → ψKS. It is then possible to write down exact analytical expressions for \(\bar{\rho}\) and \(\bar{\eta}\) as simple functions of the observables S and sin2β and of the penguin parameters r and ϕ. As an application clean lower bounds on \(\bar{\eta}\) and \(1-\bar{\rho}\) can be derived as functions of S and sin2β, essentially without hadronic uncertainty. Computing r and ϕ within QCD factorization yields precise determinations of \(\bar{\rho}\) and \(\bar{\eta}\) since the dependence on r and ϕ is rather weak. It is emphasized that the sensitivity to the phase ϕ enters only at second order and is extremely small for moderate values of this phase, predicted in the heavy-quark limit. Transparent analytical formulas are further given and discussed for the parameter C of direct CP-violation in B → π+π-. Predictions and uncertainties for r and ϕ in QCD factorization are examined in detail. It is pointed out that a simultaneous expansion in 1/mb and 1/N leads to interesting simplifications. At first order infrared divergences are absent, while the most important effects are retained. Independent experimental tests of the factorization framework are briefly discussed.