# $B\to\pi\pi$ Form Factors from Light-Cone Sum Rules with $B$-meson Distribution Amplitudes

## TL;DR

This paper develops light-cone sum rules with B-meson distribution amplitudes to calculate B to pi pi form factors, providing insights into semileptonic B decays and systematic corrections to B to rho transitions.

## Contribution

It introduces a new method to relate B to pi pi form factors with B-meson distribution amplitudes, extending the understanding of B decay form factors beyond rho dominance.

## Key findings

- Sum rules valid for high-energy, low-invariant-mass dipion states.
- Reproduction of B to rho form factors in the rho-dominance limit.
- Finite width and excited rho resonance effects contribute up to 20% in small dipion mass region.

## Abstract

We study $B\to\pi\pi$ form factors using QCD light-cone sum rules with $B$-meson distribution amplitudes. These form factors describe the semileptonic decay $B\to \pi\pi \ell\bar{\nu}_{\ell}$, and constitute an essential input in $B\to \pi\pi \ell^+\ell^-$ and $B\to \pi\pi\pi$ decays. We employ the correlation functions where a dipion isospin-one state is interpolated by the vector light-quark current. We obtain sum rules where convolutions of the $P$-wave $\bar{B}^0\to \pi^+\pi^0$ form factors with the time-like pion vector form factor are related to universal $B$-meson distribution amplitudes. These sum rules are valid in the kinematic regime where the dipion state has a large energy and a low invariant mass, and reproduce analytically the known light-cone sum rules for $B\to \rho$ form factors in the limit of $\rho$-dominance with zero width, thus providing a systematics for so-far-unaccounted corrections to $B\to\rho$ transitions. Using data for the pion vector form factor, we estimate finite width-effects and the contribution of excited $\rho$-resonances to the $B\to\pi\pi$ form factors. We find that these contributions amount up to $\sim 20\%$ in the small dipion mass region where they can be effectively regarded as a nonresonant ($P$-wave) background to the $B\to\rho$ transition.