QCD Factorization for $B\to \pi \pi \ell \nu$ Decays at Large Dipion Masses

Abstract

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

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