Measurements of the $\rho$ and $\eta$ parameters of the $V_{CKM}$ matrix and perspectives

A review of the current status of the Cabibbo-Kobayashi-Maskawa matrix ($V_{CKM}$) is presented and a special emphasis is put on the determination of the $\rho$ and $\eta$ parameters. From this study it follows that, in the Standard Model, the $B^0_s$-$\bar{B^0_s}$ oscillation frequency, $\Delta m_s$, has to lie, with 68% C.L., between 6.5 ps$^{-1}$ and 15 ps$^{-1}$, and is below 21 ps$^{-1}$ at 95% C.L. If the interest of measuring $\Delta m_s$ is underlined, the importance of a precise determination of the B meson decay constant, $f_B$, is also stressed. It is proposed to obtain a precise value for this parameter from an accurate measurement of $f_D$, the D meson decay constant, using results from lattice QCD to relate D and B hadrons. A future Tau-Charm factory could accomplish this task. It is also shown that from the present measurements, assuming the validity of the Standard Model, an already accurate value of $sin 2\beta~=~ 0.67 ^{+0.12}_{-0.13}$ is obtained. Sofar no constraint can be obtained on $sin 2\alpha$. The interest of having a direct measurement of $sin 2\alpha$ and $sin 2\beta$ at B-factories or at other facilities is reminded. Finally constraints on SUSY parameters, in the framework of a given model, obtained from a precise measurement of the CKM matrix elements are analyzed.