Isospin Violation in $\mathbf{B}\to \pi \pi$ Decays

Abstract

An isospin analysis of $B\to \pi \pi$ decays yields $\sin 2\alpha$, where $\alpha$ is the usual CKM angle $\alpha \equiv \arg [-V_{td}{V}_{tb}^{\ast }/(V_{ud}{V}_{ub}^{\ast })]$ without hadronic uncertainty if isospin is a perfect symmetry. Yet isospin symmetry is broken not only by electroweak effects but also by the $u$ and $d$ quark mass difference --- the latter drives ${\pi }^0-\eta ,{\eta }^{\prime }$ mixing and converts the isospin-perfect triangle relation between the $B\to \pi \pi$ amplitudes to a quadrilateral. The error incurred in $\sin 2\alpha$ through the neglect of the resulting isospin-violating effects can be significant, particularly if $\sin 2\alpha$ is small.

7 pages, REVTeX, epsf, 1 eps figure, talk at DPF'99