How Isospin Violation Mocks ``New'' Physics: ${\pi }^0-\eta ,{\eta }^{\prime }$ Mixing in $B\to \pi \pi$ 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. Isospin, however, is broken not only by electroweak effects but also by the u and d quark mass difference. The latter generates ${\pi }^0-\eta ,{\eta }^{\prime }$ mixing and converts the isospin-perfect triangle relation between the $B\to \pi \pi$ amplitudes to a quadrilateral. The combined isospin-violating effects impact the extracted value of $\sin 2\alpha$ in a significant manner, particularly if the latter is small.

Comments: 9 pages, 1 figure, REVTeX, epsfig; major revisions; num. error corrected