Unitarity triangles and characteristic measurables of the $3\times 3$ Cabibbo-Kobayashi-Maskawa (CKM) matrix are discussed. Beyond the $3\times 3$ CKM scheme we make a rephasing-invariant generalization of the Gronau-Wyler-Dunietz approach to determine the weak phase shift in $B^{ \pm}_u\rightarrow D K^{ \pm}$ or $B^0_d\rightarrow D K^{ *0}$ vs $\bar{ B}^0_d\rightarrow D \bar{ K}^{ *0}$, which is only sensitive to the underlying new physics in $D^0-\bar{ D}^0$ mixing. We also show that the weak angle $\gamma\equiv \arg (-V^*_{ ub}V^*_{ cd}V_{ ud}V_{ cb})$ is possible to be determined from the $CP$ asymmetries of some $B_d$ decays, even in a non-standard model with an extended quark sector. Brief comments are given on tests of unitarity of the $3\times 3$ CKM matrix.
Categories: hep-ph nucl-th