Branching ratios for $B \to K^* \gamma$ and $B \to \rho \gamma$ decays in next-to-leading order in the large energy effective theory

We calculate the so-called hard spectator corrections in ${\cal O} (\alpha_s)$ in the leading-twist approximation to the decay widths for $B \to K^{*} \gamma$ and $B \to \rho \gamma$ decays and their charge conjugates, using the Large Energy Effective Theory (LEET) techniques. Combined with the hard vertex and annihilation contributions, they are used to compute the branching ratios for these decays in the next-to-leading order (NLO) in the strong coupling $\alpha_s$ and in leading power in $\Lambda_{\rm QCD}/M_B$. These corrections are found to be large, leading to the inference that the theoretical branching ratios for the decays $B \to K^* \gamma$ in the LEET approach can be reconciled with current data only for significantly lower values of the form factors than their estimates in the QCD sum rule and Lattice QCD approaches. However, the form factor related uncertainties mostly cancel in the ratios ${\cal B}(B \to \rho \gamma)/{\cal B}(B \to K^* \gamma)$ and $\Delta = (\Delta^{+0}+ \Delta^{-0})/2$, where $\Delta^{\pm 0} = \Gamma (B^\pm \to \rho^\pm \gamma)/ [2 \Gamma (B^0 (\bar B^0)\to \rho^0 \gamma)] - 1$, and hence their measurements will provide quantitative information on the standard model parameters, in particular the ratio of the CKM matrix elements $| V_{td}/V_{ts}|$ and the inner angle $\alpha$ of the CKM-unitarity triangle. We also calculate direct CP asymmetries for the decays $B^\pm \to \rho^\pm \gamma$ and $B^0/\bar B^0 \to \rho^0 \gamma$ and find, in conformity with the observations made in the existing literature, that the hard spectator contributions significantly reduce the asymmetries arising from the vertex corrections