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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 $B^+ \to π^+ π^+ π^-$. 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 $B \to π$ 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 $B \to ππ$ form factor and the two-pion distribution amplitude. We present the set-up for both regions to leading order in both $α_s$ and $Λ_{\rm QCD} / m_b$ 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 $B\toρπ$ 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.