Measuring $|V_{\mathrm{ub}}|$ with $\overrightarrow{B}{D}_s^{+}{X}_u$ transitions

We propose the determination of the CKM matrix element $|V_{\mathrm{ub}}|$ by the measurement of the spectrum of $\overrightarrow{B}{D}_s^{+}{X}_u,$ dominated by the spectator quark model mechanism $\overline{b}\to D_s^{\left(*\right)+}ū.$ The interest in considering $\overrightarrow{B}{D}_s^{+}{X}_u$ versus the semileptonic decay is that more than 50% of the spectrum for $\overrightarrow{B}{D}_s^{+}{X}_u$ occurs above the kinematical limit for $\overrightarrow{B}{D}_s^{+}{X}_c,$ while most of the spectrum $\overrightarrow{B}l\nu X_u$ occurs below the $\overrightarrow{B}l\nu X_c$ one. Furthermore, the measure of the hadronic mass $M_X$ is easier in the presence of an identified $D_s$ than when a $\nu$ has been produced. As a consistency check, we point out that the rate $\overline{b}\to D_s^{\left(*\right)+}\overline{c}$ (including QCD corrections) is consistent with the measured $\mathrm{BR}\left(\overrightarrow{B}{D}_s^{\pm }X\right)$ by CLEO. Although the hadronic complications may be more severe in the decay mode that we propose than in the semileptonic inclusive decay, the end of the spectrum in $\overrightarrow{B}l\nu X_u$ is not well understood on theoretical grounds. We argue that, in our case, the excited mesons $D_s^{**},$ decaying into $DK,$ do not contribute and, if there is tagging of the B meson, the other mechanisms to produce a $D_s$ of the right sign are presumably small, of ${O(10\mathrm{}}^{-2})$ relative to the spectator amplitude, or can be controlled by kinematical cuts. We discuss in detail the hadronic uncertainties of the method and present an error calculation on $|V_{\mathrm{qb}}|$ $\mathrm{}(q=u,c)\mathrm{}$ and on the ratio