Measuring |V_{td} / V_{ub}| through B -> M ν\barν (M=π,K,ρ,K^*) decays

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