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 ${\cal O}(\lambda ^2)$ 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 - ${\cal O}(\lambda ^2)$ and ${\cal O}(\lambda ^4)$ thus generating an asymmetry of a few percent and of about 0.1 %, respectively. The former can be measured in $B_s \to \psi \eta, \psi \phi$ {\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.