CLEO measurement of $B \to \pi^+ \pi^-$ and determination of weak phase $\alpha$

Abstract

The CLEO collaboration has recently reported a (first) measurement of $BR (B \to \pi^+ \pi^-) = (4.7^{+1.8}_{-1.5} \pm 0.6) \times 10^{-6}$. We study, using recent results on QCD improved factorization, the implications of this measurement for the determination of the CKM phase $\alpha$ and also for the rate for $B \to \pi^0 \pi^0$. If the $B \to \pi^-$ form factor is large $(\stackrel{>}{\sim} 0.3)$, then we find that the CLEO measurement favors small $|V_{ub}/V_{cb}|$ so that the expected error (due to neglecting the QCD penguin amplitude) in the measurement of $\alpha$ using {\em only} the time-dependent analysis of the decay $B \to \pi^+ \pi^-$ is large $\sim 15^{\circ}$. However, if $|V_{ub}/V_{cb}|$ is known, then it is possible to determine the correct value of $\sin 2 \alpha$.