Form factors in $B\to \pi\pi\ell\bar\nu_\ell$ from QCD light-cone sum rules

TL;DR

This paper calculates $B o ho o ext{dipion}$ form factors using QCD light-cone sum rules, revealing the dominance of the lowest partial wave and the limitations of the $ ho$-meson approximation.

Contribution

It introduces a method to derive form factors from QCD light-cone sum rules considering dipion distribution amplitudes, addressing kinematic singularities and partial wave contributions.

Findings

Form factors are calculated for $0<q^2\, extless\,12$ GeV$^2$

Higher partial waves are suppressed compared to the P-wave

The $ ho$-meson does not fully saturate the P-wave contribution

Abstract

The form factors of the semileptonic $B\to \pi\pi\ell\bar\nu$ decay are calculated from QCD light-cone sum rules with the distribution amplitudes of dipion states. This method is valid in the kinematical region, where the hadronic dipion state has a small invariant mass and simultaneously a large recoil. The derivation of the sum rules is complicated by the presence of an additional variable related to the angle between the two pions. In particular, we realize that not all invariant amplitudes in the underlying correlation function can be used, some of them generating kinematical singularities in the dispersion relation. The two sum rules that are free from these ambiguities are obtained in the leading twist-2 approximation, predicting the $\bar{B}^0\to \pi^+\pi^0$ form factors $F_{\perp}$ and $F_{\parallel}$ of the vector and axial $b\to u$ current, respectively. We calculate these form factors at the momentum transfers $0<q^2\lesssim 12 $ GeV$^2$ and at the dipion mass close to the threshold $4m_\pi^2$. The sum rule results indicate that the contributions of the higher partial waves to the form factors are suppressed with respect to the lowest $P$-wave contribution and that the latter is not completely saturated by the $\rho$-meson term.