Implications of the Fleischer-Mannel Bound

Fleischer and Mannel (FM) have shown that it may become possible to constrain the angle $\gamma$ of the unitarity triangle from measurements of various $B\to\pi K$ decays. This constraint is independent of hadronic uncertainties to the few percent level. We show that, within the Standard Model, the FM bound gives strong constraints on the CKM parameters. In particular, it could predict a well defined sign for $\sin2\gamma$ and $\sin2\alpha$. In a class of extensions of the Standard Model, where the New Physics affects only $\Delta B=2$ (and, in particular, not $\Delta B=1$) processes, the FM bound can lead to constraints on CP asymmetries in $B$ decays into final CP eigenstates even if $B-\bar B$ mixing is dominated by unknown New Physics. In our analysis, we use a new method to combine in a statistically meaningful way the various measurements that involve CKM parameters.