Isospin Violation in B! Decays

An isospin analysis of B ! decays yields sin 2 , where is the usual CKM angle arg[ VtdV tb=(VudV ub)] without hadronic uncertainty if isospin is a perfect symmetry. Yet isospin symmetry is broken not only by electroweak e ects but also by the u and d quark mass di erence | the latter drives 0 ; 0 mixing and converts the isospin-perfect triangle relation between the B ! amplitudes to a quadrilateral. The error incurred in sin 2 through the neglect of the resulting isospin-violating e ects can be signi cant, particularly if sin 2 is small.