CP Violation in B -> pi+ pi- and the Unitarity Triangle

Abstract

We analyze the extraction of weak phases from CP violation in $B\toπ^+π^-$ decays. We propose to determine the unitarity triangle $(\barρ,\barη)$ by combining the information on mixing induced CP violation in $B\toπ^+π^-$, $S$, with the precision observable $\sin 2β$ obtained from the CP asymmetry in $B\toψK_S$. It is then possible to write down exact analytical expressions for $\barρ$ and $\barη$ as simple functions of the observables $S$ and $\sin 2β$, and of the penguin parameters $r$ and $ϕ$. As an application clean lower bounds on $\barη$ and $1-\barρ$ can be derived as functions of $S$ and $\sin 2β$, essentially without hadronic uncertainty. Computing $r$ and $ϕ$ within QCD factorization yields precise determinations of $\barρ$ and $\barη$ 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 $B\toπ^+π^-$. 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 $ϕ$ 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.