On the Other Five KM Triangles

Abstract

A comprehensive program of \cp~studies in heavy flavour decays has to go beyond observing large \cp asymmetries in nonleptonic B decays and finding that the sum of the three angles of the KM triangle is consistent with 180${}^{\circ }$. There are many more correlations between observables encoded in the KM matrix; those can be expressed through five KM triangles in addition to the one usually considered. To test the completeness of the KM description one has to obtain a highly overconstrained data set sensitive to $\mathcal{O}({\lambda }^2)$ effects with $\lambda =\sin {\theta }_C$. Those fall into two categories: (i) Certain large angles agree to leading order only, yet differ in order ${\lambda }^2$ in a characteristic way. (ii) Two observables angles are - for reasons specific to the KM ansatz - $\mathcal{O}({\lambda }^2)$ and $\mathcal{O}({\lambda }^4)$ thus generating an asymmetry of a few percent and of about 0.1 %, respectively. The former can be measured in $B_s\to \psi \eta ,\psi \varphi$ {\em without} hadronic uncertainty, the latter in Cabibbo suppressed D decays. The intervention of New Physics could boost these effects by an order of magnitude. A special case is provided by $D^{+}\to K_{S,L}{\pi }^{+}$ vs. $D^{-}\to K_{S,L}{\pi }^{-}$. Finally, \cp~asymmetries involving $D^0-{\bar{D}}^0$ oscillations could reach observable levels only due to New Physics.

20 pages, 1 embedded figure, LATEX; while the substance of this paper remains unchanged, the discussion has been extended in the interest of clarity and illustrated by one figure with 6 subfigures; some typos have been corrected and relevant papers added to the references