QCD analysis of $B \to ππ$ form factors and the $\vert V_{ub} \vert$ extraction from $B_{l4}$ decays

Abstract: The decays of $B_{l4}$ offer an independent method for determining the CKM matrix element $\vert V_{ub} \vert$, with a QCD analysis of the underlying $B \to \pi\pi$ form factors serving as a crucial component in this process. We present a comprehensive QCD analysis of the $B \to \pi\pi$ form factors within the framework of light-cone sum rules, employing two-pion light-cone distribution amplitudes ($2\pi$DAs). For the first time, the twist-three distribution amplitudes are explicitly defined, and their contributions are systematically calculated. Leveraging the recently measured partial decay fractions of $B^+ \to \pi^+\pi^- l^+ \nu_l$ by the Belle detector, we obtain $\vert V_{ub} \vert = \left( 4.27 \pm 0.49\vert_{\rm data} \pm 0.55\vert_{\rm LCSRs} \right) \times 10^{-3}$ within the dominant region of the $\rho$ resonance. We clarify that the relatively small value of $\vert V_{ub} \vert$ extracted from $B \to \rho l \nu$, when compared to the golden channel $B \to \pi l\nu$, arises due to the finite width and nonresonant effects. We anticipate future measurements of purely isovector and isoscalar $B_{l4}$ decays, which will aid in reducing uncertainties by enhancing our understanding of the $\pi\pi$ phase shifts in the $2\pi$DAs through heavy flavor decays.