The factorizable amplitude in $B^0\to {\pi }^{+}{\pi }^{-}$

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