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

Abstract

Analysis of the present data on the semileptonic process $B \to \pi \ell \nu$ 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 $q^2 = m_\pi^2$ is known to be related to the factorized color-favored (``$T$'', 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 $\alpha = \phi_2$ 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.