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We introduce a factorization formula for semi-leptonic $b \to u$ transitionsin the exclusive decay mode $B^- \to \pi^+\pi^-\ell^-\bar\nu_\ell$ in the limitof large pion energies and large dipion invariant mass. One contribution can bedescribed in terms of a universal $B \to \pi$ form factor and the convolutionof a short-distance kernel $T^{\rm I}$ with the respective light-conedistribution amplitudes (LCDAs) of the positively charged pion. The secondcontribution, at leading power, completely factorizes, with a short-distancekernel $T^{\rm II}$ convoluted with the leading-twist LCDAs for both pions andthe $B$-meson. We calculate the leading contributions to the short-distancekernels $T^{\rm I}$ and $T^{\rm II}$ in fixed-order perturbation theory, anddiscuss the approximate relations among the resulting $B \to \pi\pi$partial-wave form factors. Our results provide useful theoretical constraintsfor phenomenological models that aim to analyze the complete $B \to \pi\pi\ell\nu$ phase space.

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