Next-to-Next-to-Leading-Order Corrections to the $B \to \pi $ Form Factors from Light-Cone Sum Rules

Abstract

By incorporating the available leading-power results at $\mathcal{O}(\alpha_s)$ and next-to-leading-power corrections at tree level, we improve the precision of the theoretical predictions for $B\to\pi$ form factors to the $\mathcal{O}(\alpha_s^2\beta_0)$ level in the large-recoil region using the light-cone sum rule approach with $B$-meson light-cone distribution amplitudes. We find that the QCD corrections at $\mathcal{O}(\alpha_s^2\beta_0)$ contribute approximately $+6.1\%$ compared to the tree-level result. Combining the light-cone sum rule predictions in the large-recoil region, lattice QCD results in the small-recoil region, we perform a combined fit for the $B\to\pi$ form factors across the full kinematic range. Utilizing these form factors, we calculate the branching ratios, lepton-flavor-universality ratio $R_\pi$, forward-backward asymmetry $\mathcal{A}_{\rm FB}$, flat term $\mathcal{F}_{\rm H}$ and polarization asymmetry $\mathcal{A}_{\rm \lambda_\ell}$ of $B\to\pi\mu\bar{\nu}_{\mu}$ and $B\to\pi\tau\bar{\nu}_{\tau}$ decays. Using the experimentally measured $q^2$-binned differential branching ratios of $B\to\pi\mu\bar{\nu}_{\mu}$ decay as input, employing the Bourrely-Caprini-Lellouch parametrization, we extract the Cabibbo-Kobayashi-Maskawa matrix element $|V_{ub}| = 3.73(14) \times 10^{-3}$.