Information on $B\to \pi \pi$ Provided by the Semileptonic Process $B\to \pi l\nu$

Analysis of the present data on the semileptonic process $B\to \pi \mathrm{\ell }\nu$ indicates that they have not yet reached the precision to provide adequate information on the $B\to \pi$ form factor $F_{+}(q^2)$, which for $q^2 = m_\pi^2$isknowntoberelatedtothefactorizedcolor-favored(\mathrm{‘}\mathrm{‘}$T$‘’, or tree'') contribution to $B^0 \to \pi^+ \pi^-$. It is shown here that with around 500 $B \to \pi \ell \nu$ events in which rate and spectrum are measured one can improve the accuracy of $T$ by a significant amount. A recent CLEO determination of the $D^* D \pi$ coupling constant is compared with an earlier prediction, and its role in the description of the $B \to \pi$ form factors is noted. When combined with an estimate of the penguin amplitude ($P$’') obtained using flavor SU(3) symmetry from $B\to K\pi$ decays, information on $T$ allows one to gauge the effects of the penguin amplitude on extraction of the weak phase $\alpha ={\varphi }_2$ from the time-dependent CP-violating rate asymmetry in $B^0\to {\pi }^{+}{\pi }^{-}$. The constraint on $\alpha$ implied by a recent experimental result on this asymmetry is described.