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

Abstract

The $B_{l4}$ decays offer an independent method for determining the Cabibbo-Kobayashi-Maskawa matrix element $|V_{ub}|$, with a QCD analysis of the underlying $B\to \pi \pi$ form factors serving as a crucial component. 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. For the first time, we derive the twist-three $2\pi$DAs explicitly by exploiting the conformal symmetry of massless QCD. This breakthrough enables the first systematic calculation of higher-power corrections to the $B\to \pi \pi$-type transitions. Leveraging the recently measured partial decay fractions of $B^{+}\to {\pi }^{+}{\pi }^{-}{l}^{+}{\nu }_l$ by the Belle detector, we obtain $|V_{ub}|=\left(4.27\pm 0.49{|}_{\mathrm{data}}\pm 0.55{|}_{\mathrm{LCSRs}}\right)\times 10^{-3}$ within the dominant region of the $\rho$ resonance. We clarify that the relatively small value of $|V_{ub}|$ 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 two-pion distribution amplitudes through heavy flavor decays.

Comments: 6+4 pages, 2 figures, 1 table, version accepted for publication as a Letter in Physical Review D