Heavy-to-light Structure Functions at $\mathcal{O}(\alpha_s^3)$ in QCD

We present the first complete $\mathcal{O}(\alpha_s^2)$ and $\mathcal{O}(\alpha_s^3)$ perturbative QCD corrections to all five heavy-to-light structure functions underlying the triple-differential semi-leptonic decay rates of heavy quarks. This is achieved via a hybrid computational strategy that combines an efficient linear interpolation (with a suitable function basis) based on stratified Gauss-Kronrod points in the leptonic-mass $q^2$ with the differential equations in the other variable, further armed with reduced numerical $\varepsilon$-dependence. Among the selected applications, we highlight the state-of-the-art prediction $\Gamma(B \rightarrow X_u \ell \bar{\nu}_{\ell}) = \frac{|V_{ub}|^2}{|3.82\times 10^{-3}|^2}\,\big( 6.53 \,\pm 0.12 \, \pm 0.13\, \pm 0.03\, \big) \times 10^{-16}\,\text{GeV}\,$ derived in the kinetic-mass scheme. We report several notable observations regarding the convergence of the first three orders of QCD corrections to the $q^2$-spectrum and to inclusive moments of the lepton-energy spectrum in semi-leptonic weak decays of $b$- and $c$-quark in different quark-mass schemes; they are important both for improving the inclusive determinations of the relevant CKM elements, non-perturbative dynamical parameters, and for gaining new insights into the potential impact of high-order QCD corrections. Lastly we discuss a novel interesting point encountered in the consistent perturbative reformulation of the differential $q^2$-spectrum from the pole-mass to other mass schemes: certain boundary-effect terms are identified that are non-vanishing for $b \rightarrow u \ell \bar{\nu}_{\ell}$ firstly at $\mathcal{O}(\alpha_s^3)$; their incorporation is essential to preserve the integrity of the integrated moments of the perturbatively re-expanded $q^2$-spectrum but necessitates histogramming from $\mathcal{O}(\alpha_s^3)$ onward even within pure perturbation theory.