Three-body non-leptonic B decays and QCD factorization

We extend the framework of QCD factorization to non-leptonic B   decays into three light mesons, taking as an example the decay B+→π+π+π−B+→π+π+π−. We discuss the factorization properties of this decay in different regions of phase space. We argue that, in the limit of very large b  -quark mass, the central region of the Dalitz plot can be described in terms of the B→πB→π form factor and the B and π   light-cone distribution amplitudes. The edges of the Dalitz plot, on the other hand, require different non-perturbative input: the B→ππB→ππ form factor and the two-pion distribution amplitude. We present the set-up for both regions to leading order in both αsαs and ΛQCD/mbΛQCD/mb and discuss how well the two descriptions merge. We argue that for realistic B  -meson masses there is no perturbative center in the Dalitz plot, but that a systematic description might be possible in the context of two-pion states. As an example, we estimate the B→ρπB→ρπ branching fraction beyond the quasi-particle approximation. We also discuss the prospects for studies of three-body and quasi-two-body non-leptonic B decays from QCD.