CP Violation in B -> pi+ pi- and the Unitarity Triangle
  We analyze the extraction of weak phases from CP violation in $B\to\pi^+\pi^-$ decays. We propose to determine the unitarity triangle $(\bar\rho,\bar\eta)$ by combining the information on mixing induced CP violation in $B\to\pi^+\pi^-$, $S$, with the precision observable $\sin 2\beta$ obtained from the CP asymmetry in $B\to\psi K_S$. It is then possible to write down exact analytical expressions for $\bar\rho$ and $\bar\eta$ as simple functions of the observables $S$ and $\sin 2\beta$, and of the penguin parameters $r$ and $\phi$. As an application clean lower bounds on $\bar\eta$ and $1-\bar\rho$ can be derived as functions of $S$ and $\sin 2\beta$, essentially without hadronic uncertainty. Computing $r$ and $\phi$ within QCD factorization yields precise determinations of $\bar\rho$ and $\bar\eta$ since the dependence on $r$ and $\phi$ is rather weak. It is emphasized that the sensitivity to the phase $\phi$ 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 $B\to\pi^+\pi^-$. We also discuss alternative ways to analyze $S$ and $C$ that can be useful if new physics affects $B_d$--$\bar B_d$ mixing. Predictions and uncertainties for $r$ and $\phi$ in QCD factorization are examined in detail. It is pointed out that a simultaneous expansion in $1/m_b$ 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. 
Gerhard Buchalla and A. Salim Safir (LMU Munich)