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

TL;DR

The authors compute next-to-next-to-leading-order corrections in the large-β0 limit to the B→π form factors within a B-meson light-cone sum-rule framework, achieving NNLO accuracy in the large-recoil region by evaluating two-loop fermion-loop contributions and constructing NNLL resummed expressions. They confirm factorization-scale independence, verify the Wandzura–Wilczek relation at this order, and combine LCSR results with lattice QCD to obtain form factors across the full kinematic range via a BCL z-series fit. These form factors enable predictions of branching fractions, R_π, and angular observables for B→πℓν decays, and, by fitting to experimental data, yield a precise |V_{ub}| = 3.73(14) × 10^{-3}. The results reduce theoretical uncertainties and provide a consistent SM benchmark for lepton-flavor universality tests and future searches for new physics in semileptonic B decays.

Abstract

By incorporating the available leading-power results at $\mathcal{O}(α_s)$ and next-to-leading-power corrections at tree level, we improve the precision of the theoretical predictions for $B\toπ$ form factors to the $\mathcal{O}(α_s^2β_0)$ 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 $\mathcal{O}(α_s^2β_0)$ 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 $B\toπ$ form factors across the full kinematic range. Utilizing these form factors, we calculate the branching ratios, lepton-flavor-universality ratio $R_π$, forward-backward asymmetry $\mathcal{A}_{\rm FB}$, flat term $\mathcal{F}_{\rm H}$ and polarization asymmetry $\mathcal{A}_{\rm λ_\ell}$ of $B\toπμ\barν_μ$ and $B\toπτ\barν_τ$ decays. Using the experimentally measured $q^2$-binned differential branching ratios of $B\toπμ\barν_μ$ decay as input, employing the Bourrely-Caprini-Lellouch parametrization, we extract the Cabibbo-Kobayashi-Maskawa matrix element $|V_{ub}| = 3.73(14) \times 10^{-3}$.

Figures (8)

• Figure 1: Diagrammatical representation of the correlation function $\Pi_\mu(n\cdot p, \bar{n}\cdot p)$ at tree level.
• Figure 2: QCD corrections for the correlation function $\Pi_\mu(n\cdot p,\bar{n}\cdot p)$ at $\mathcal{O}(\beta_0 \alpha_s^2)$.
• Figure 3: Comparison of the leading logarithm resummation improved tree-level contribution, next-to-leading logarithm resummation improved one-loop correction, next-to-leading power correction at tree level and total result to the vector form factor $f_{B\pi}^+$ (left panel) and scalar form factor $f_{B\pi}^0$ (right panel) in the kinematic region of $0\leq q^2\leq 8~\rm{GeV}^2$. The shaded bands represent the uncertainties from the variation of hard scale $\mu_{h}$ and factorization scale $\mu$.
• Figure 4: The theory prediction for the vector and scalar form factors of $B\to\pi$ transition versus $z$ (left panel) and versus $q^2$ (right panel) obtained from the combined fit of the updated LCSR (from this work) and the LQCD (from Flynn:2015mhaFermilabLattice:2015mwy) data points in the entire kinematic region. An alternative $z$-series fit for the form factors exclusively using the "only LQCD" data points is presented for a comparison.
• Figure 5: Theoretical predictions for the differential decay width of the $B\to\pi\mu\bar{\nu}_\mu$ (left) and $B\to\pi\tau\bar{\nu}_\tau$ (right) decay in the entire $q^2$ region, with or without LCSR as input. The experimental measurements from the BaBar BaBar:2010efpBaBar:2012thb, Belle Belle:2010hepBelle:2013hlo, and Belle II Belle-II:2022imn collaborations for the decay $B\to\pi\mu\bar{\nu}_\mu$ are displayed for a comparison.