The factorizable amplitude in $B^0 \to π^+ π^-$

Using the measured spectrum shape for $B \to π\ell ν$, the rate for $B^+ \to π^+ π^0$, information on the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $|V_{ub}|$, and theoretical inputs from factorization and lattice gauge theory, we obtain an improved estimate of the ``tree'' contribution to $B^0 \to π^+ π^-$. We find the branching ratio $\b(B^0 \to π^+ π^-)|_{\rm tree} = (5.25^{+1.67}_{-0.50}) \times 10^{-6}$, to be compared with the experimental value $\b(B^0 \to π^+ π^-) = (4.55 \pm 0.44) \times 10^{-6}$. The fit implies $|V_{ub}| = (3.62 \pm 0.34) \times 10^{-3}$. Implications for tree-penguin interference in $B^0 \to π^+ π^-$ and for other charmless $B$ decays are discussed.

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