Isospin Violation in ${\bf B\to ππ}$ Decays

Abstract

An isospin analysis of $B\to ππ$ decays yields $\sin 2α$, where $α$ is the usual CKM angle $α\equiv {\rm arg} [-V_{td} V_{tb}^\ast/(V_{ud}V_{ub}^\ast)]$ 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 $π^0-η,η'$ mixing and converts the isospin-perfect triangle relation between the $B\to ππ$ amplitudes to a quadrilateral. The error incurred in $\sin 2α$ through the neglect of the resulting isospin-violating effects can be significant, particularly if $\sin 2α$ is small.