NNLO vertex corrections to non-leptonic B decays: Tree amplitudes

The colour-suppressed tree amplitude in non-leptonic B decays is particularly sensitive to perturbative and non-perturbative corrections. We calculate the two-loop (NNLO) vertex corrections to the colour-suppressed and colour-allowed tree amplitudes in QCD factorization. Our results are given completely analytically, including the full dependence on the charm quark mass. We then update theoretical predictions for a range of interesting observables derived from ππ, πρ and ρρ   final states that do not depend (significantly) on penguin contributions, and hence are now available with NNLO accuracy. We observe good agreement with experimental data within experimental and theoretical errors, except for observables involving the π0π0π0π0 branching fraction.