Triple Differential Heavy-to-light Semi-leptonic Decays at Next-to-Next-to-Next-to-Leading Order in QCD

Abstract

We report the first complete calculation of the five heavy-to-light hadronic structure functions underlying semi-leptonic heavy-quark decays at next-to-next-to-next-to-leading order ($\mathcal{O}(\alpha_s^3)$) in perturbative QCD. This theoretical advance, achieved via an innovative hybrid computational strategy, enables precision predictions for triple differential decay rates. The results are essential for harnessing the potential of high-precision experiments at Belle II, BES III, and LHCb. Selected applications of this work include a state-of-the-art prediction for the inclusive $B \to X_u \ell \nu$ width, crucial for a percent-level determination of $|V_{ub}|$, and the first $\mathcal{O}(\alpha_s^3)$ results for lepton-energy moments in charm decays, vital for extracting $|V_{cs}|$ and $|V_{cd}|$. Our analysis also reveals significant higher-order corrections in the large-$q^2$ region of $b \to u$ transitions, offering new insights into the persistent tension between inclusive and exclusive $|V_{ub}|$ determinations.