QCD factorization for B → ππℓν decays at large dipion masses

We introduce a factorization formula for semi-leptonic b → u transitions in the exclusive decay mode B⁻→ π⁺π⁻⁻ν̄ in the limit of large pion energies and large dipion invariant mass. One contribution can be described in terms of a universal B → π form factor and the convolution of a short-distance kernel T I with the respective light-cone distribution amplitudes (LCDAs) of the positively charged pion. The second contribution, at leading power, completely factorizes, with a short-distance kernel T II convoluted with the leading-twist LCDAs for both pions and the B-meson. We calculate the leading contributions to the short-distance kernels T I and T II in fixed-order perturbation theory, and discuss the approximate relations among the resulting B → ππ partial-wave form factors. Our results provide useful theoretical constraints for phenomenological models that aim to analyze the complete B → ππℓν phase space.