Anomalous dimension matrix for radiative and rare semileptonic B decays up to three loops.

We compute the complete O(αs2) anomalous dimension matrix relevant for the b→sγ, b→sg and b→sℓ+ℓ− transitions in the standard model and some of its extensions. For radiative decays we confirm the results of Misiak and Münz, and of Chetyrkin, Misiak and Münz. The O(αs2) mixing of four-quark into semileptonic operators is instead a new result and represents one of the last missing ingredients of the next-to-next-to-leading-order analysis of rare semileptonic B meson decays.