QCD Factorization for $B \to ππ\ellν$ Decays at Large Dipion Masses

Abstract

We introduce a factorization formula for semi-leptonic $b \to u$ transitions in the exclusive decay mode $B^- \to π^+π^-\ell^-\barν_\ell$ in the limit of large pion energies and large dipion invariant mass. One contribution can be described in terms of a universal $B \to π$ form factor and the convolution of a short-distance kernel $T^{\rm I}$ 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 $T^{\rm II}$ convoluted with the leading-twist LCDAs for both pions and the $B$-meson. We calculate the leading contributions to the short-distance kernels $T^{\rm I}$ and $T^{\rm II}$ in fixed-order perturbation theory, and discuss the approximate relations among the resulting $B \to ππ$ partial-wave form factors. Our results provide useful theoretical constraints for phenomenological models that aim to analyze the complete $B \to ππ\ellν$ phase space.

38 pages, 5 figures; v2: as published, conclusions unchanged