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Exclusive non-leptonic two-body decays of $B$ mesons have been studiedextensively in the past two decades within the framework of factorization.However, the exploration of the corresponding three-body case has only startedrecently, in part motivated by new data. We consider here the simplestnon-leptonic three-body $B$ decays from the point of view of factorization,namely heavy-to-heavy transitions. We provide a careful derivation of theSCET/QCDF factorized amplitudes to NNLO in $\alpha_s$, and discuss thenumerical impact of NLO and NNLO corrections. We then study the narrow-widthlimit, showing that the three-body amplitude reproduces analytically the knownquasi-two-body decay amplitudes, and compute finite-width corrections. Finally,we discuss certain observables that are sensitive to perturbative NLO and NNLOcorrections and to higher Gegenbauer moments of the dimeson LCDAs. This is thefirst study of non-leptonic three-body $B$ decays to NNLO in QCD.

Three-Body Non-Leptonic Heavy-to-heavy $B$ Decays at NNLO in QCD