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

TL;DR

This paper develops a QCD factorization framework for semi-leptonic B decays into two pions at large dipion masses, enabling better theoretical understanding and modeling of these processes.

Contribution

It introduces a new factorization formula for $B o ho ho$ decays at large dipion masses, including calculations of short-distance kernels and relations among form factors.

Findings

Derived factorization formulas for $B o ho ho$ decays.

Calculated leading-order short-distance kernels $T^{ m I}$ and $T^{ m II}$.

Provided theoretical constraints for phenomenological models.

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 $T^{\rm I}$ 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 $B \to \pi\pi$ 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.