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 $B^+ \to \pi^+ \pi^+ \pi^-$. 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 $B \to \pi$ form factor and the $B$ and $\pi$ light-cone distribution amplitudes. The edges of the Dalitz plot, on the other hand, require different non-perturbative input: the $B \to \pi \pi$ form factor and the two-pion distribution amplitude. We present the set-up for both regions to leading order in both $\alpha_s$ and $\Lambda_{\rm QCD} / m_b$ 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 $B\to\rho\pi$ 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.