Maximum likelihood analysis for the Cabibbo-Kobayashi-Maskawa matrix

The probability of possible values for the four angles of the CKM matrix are determined by finding the ${\chi }^2$ values for each choice of angles fitting to ten standard experiments. The ${\chi }^2$ isosurface plots are first displayed in the space of three of the CKM angles showing the dependence of the bounds of $\delta$ and greatly improving the lower bound on $V_{\mathrm{cb}}$. The maximum likelihood ${\chi }^2$ contour plot in the $\rho -\eta$ plane shows the probability of each location for the vertex of the unitarity triangle which controls $\mathrm{CP}$-violating asymmetries in $B^0$ decays. We find for $m_t=150\mathrm{}$ GeV that at the $2\sigma$ level the signal for asymmetries measuring $sin\left(2\mathrm{}\beta \right)$ is at least $sin\left(2\mathrm{}\beta \right)\ge 0.25$. At the $1\sigma$ level the signal is at least $sin\left(2\mathrm{}\beta \right)\ge 0.40$, with the preferred range of $sin\left(2\mathrm{}\beta \right)=0.6\mathrm{}\pm 0.2$.