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

Abstract

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 $\varphi$. 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 $\varphi$ within QCD factorization yields precise determinations of $\bar{\rho }$ and $\bar{\eta }$ since the dependence on $r$ and $\varphi$ is rather weak. It is emphasized that the sensitivity to the phase $\varphi$ 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 $\varphi$ 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.

Comments: 26 pages, 7 figures