Complete electroweak matching for radiative B decays

We compute the complete two–loop O(α) Wilson coefficients relevant for radiative decays of the B meson in the SM. This is a necessary step in the calculation of the O(αα s ln n mb/MW ) corrections and improves on our previous analysis of electroweak effects in B → Xsγ. We describe in detail several interesting technical aspects of the calculation and include all dominant QED matrix elements. In our final result, we neglect only terms originated from the unknown O(ααs) evolution of the Wilson coefficients and some suppressed two–loop matrix elements. Due to the compensation among different effects, we find that non–trivial electroweak corrections decrease the branching ratio by about 3.6% for a light Higgs boson, very close to our previous result. The corresponding up–to–date SM prediction for the branching ratio with Eγ > 1.6GeV is (3.61± 0.30)× 10.