We propose a method of extracting the Cabibbo-Kobayashi-Maskawa matrix element $|V_{ cb}|$ from two-body hadronic decay processes of $B\to DK$ with precisely determined form factors of $B$ meson semi-leptonic decays. The amplitude $\mathcal{ M}(\bar{ B}^0 \to D^+ K^-)$ which does not include the effect of hadronic final state interactions can be theoretically evaluated by using factorization and form factors of semi-leptonic B decays. We can obtain all the amplitudes in an isospin relation $\mathcal{ A}(B^-\to D^0K^-) =\mathcal{ A}(\bar{ B}^0\to D^+K^{ -})+\mathcal{ A}(\bar{ B}^0\to D^{ 0}\bar{ K}^0)$ including the effect of hadronic final state interactions as well as $|V_{ cb}|$ using the experimental data of branching fractions of these three processes with a truncation of the states which contribute to the hadronic final state interactions. The extracted value of $|V_{ cb}|$ is $(37\pm 6)\times 10^{ -3}$. The decay processes of $B\to DK^{ *}$ and $B\to D^{ *}K$ can also be used in the same way and the extracted values of $|V_{ cb}|$ are $(41\pm 7)\times 10^{ -3}$ and $(42\pm 9)\times 10^{ -3}$, respectively. This method becomes possible by virtue of recent precise determinations of the form factors of semi-leptonic B decays. The uncertainties of $|V_{ cb}|$ by this method are expected to be reduced by the results of future B-factory experiments and lattice calculations.
Categories: hep-ph