Measuring $|V_{\mathrm{td}}{/V}_{\mathrm{ub}}|$ $(=\sin \gamma /\sin \beta$ within the SM) through $\overrightarrow{B}M\nu \overline{\nu }\hspace{0.5em}(M=\pi ,K,\rho {,K}^{*})$ decays

We propose a new method for the precise determination of $|V_{\mathrm{td}}{/V}_{\mathrm{ub}}|$ from the ratios of branching ratios $\mathcal{B}\left(\overrightarrow{B}\rho \nu \overline{\nu }\right)/\mathcal{B}\left(\overrightarrow{B}\rho l\nu \right)$ and $\mathcal{B}\left(\overrightarrow{B}\pi \nu \overline{\nu }\right)/\mathcal{B}\left(\overrightarrow{B}\pi l\nu \right).\mathrm{}$ These ratios depend only on the ratio of the Cabibbo-Kobayashi-Maskawa (CKM) elements $|V_{\mathrm{td}}{/V}_{\mathrm{ub}}|$ with little theoretical uncertainty, when very small isospin breaking effects are neglected. As is well known, $|V_{\mathrm{td}}{/V}_{\mathrm{ub}}|$ equals $(\sin \gamma /\sin \beta )$ for the CKM version of $\mathrm{CP}$ violation within the standard model. We also give in detail analytical and numerical results on the differential decay width $d\Gamma (\overrightarrow{B}{K}^{*}\nu \overline{\nu }{)/dq}^2$ and the ratio of the differential rates $d\mathcal{B}(\overrightarrow{B}\rho \nu \overline{\nu }{)/dq}^2/d\mathcal{B}(\overrightarrow{B}{K}^{*}\nu \overline{\nu }{)/dq}^2$ as well as $\mathcal{B}\left(\overrightarrow{B}\rho \nu \overline{\nu }\right)/\mathcal{B}\left(\overrightarrow{B}{K}^{*}\nu \overline{\nu }\right)$ and $\mathcal{B}\left(\overrightarrow{B}\pi \nu \overline{\nu }\right)/\mathcal{B}\left(\overrightarrow{B}K\nu \overline{\nu }\right).\mathrm{}$