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

TL;DR

The paper delivers 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 underpinning triple-differential semi-leptonic decays, using a hybrid GK-interpolation and differential-equation approach to master integrals with controlled $\varepsilon$-dependence. By applying these results to $t\to bW$, $B\to X_u\ell\bar\nu_{\ell}$, and $c\to q\ell\bar\nu_{\ell}$ observables, the work demonstrates improved convergence and reduced scale uncertainties across multiple mass schemes (OS, $\overline{\mathrm{MS}}$, kinetic, 1S, and $\sigma$-mass). A key technical insight is the identification of boundary-effect terms that emerge when reformulating differential spectra in short-distance mass schemes, necessitating histogram-based formulation for consistent perturbative moments, particularly in $b\to u$ decays. The results yield state-of-the-art predictions for inclusive widths, differential spectra, and lepton-energy moments, with direct implications for precise extractions of CKM elements such as $|V_{ub}|$ and improved understanding of nonperturbative HQET parameters. Taken together, these advances sharpen the theoretical toolkit for heavy-flavor phenomenology and guide future analyses at Belle II, LHCb, and future colliders.

Abstract

We present the first complete $\mathcal{O}(α_s^2)$ and $\mathcal{O}(α_s^3)$ 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 $Γ(B \rightarrow X_u \ell \barν_{\ell}) = \frac{|V_{ub}|^2}{|3.82\times 10^{-3}|^2}\,\big( 6.53 \,\pm 0.12 \, \pm 0.13\, \pm 0.03\, \big) \times 10^{-16}\,\text{GeV}\,$ 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 $b \rightarrow u \ell \barν_{\ell}$ firstly at $\mathcal{O}(α_s^3)$; their incorporation is essential to preserve the integrity of the integrated moments of the perturbatively re-expanded $q^2$-spectrum but necessitates histogramming from $\mathcal{O}(α_s^3)$ onward even within pure perturbation theory.

Figures (13)

• Figure 1: The kinematics of the semi-leptonic weak decay of a heavy quark $Q$ with momentum $p$ into a pair of on-shell leptons with momentum $p_l$ and $p_v$, plus a bunch of QCD partons with the total momentum $p_X = p - p_l - p_v$.
• Figure 2: The 2-dimensional phase-space in terms of the rescaled kinematic variables $\{m_w^2/m_Q^2,\, E_w/m_Q\}$.
• Figure 3: Top row: from left to right are the plots for the numerical results of $W_1\,, W_2\,, W_3$ in a 2-dimensional regular phase-space region $R_2$, multiplied by the Jaccobian of the coordinate transformation, up to $\mathcal{O}(\alpha_s^{1})\,,\mathcal{O}(\alpha_s^{2})\,, \mathcal{O}(\alpha_s^{3})$ indicated respectively by blue, green, and red colors. Bottom row: from left to right are the plots for the numerical results of $W_4\,, W_5$ and the 2-fold distribution defined in \ref{['eq:2Dparametric']} plotted in a similar way as those in the top row.
• Figure 4: The left plot shows the perturbative coefficients after subtracting $\mathcal{O}(\alpha_s^{N+1} \beta_0^{N})$ with $N \geq 1$ at each order in $\alpha_s$, indicated respectively by blue, green, and red colors; the $\mathcal{O}(\alpha_s^{1})$-coefficient is included (in blue color) for reference. The right plot shows the ratios of the non-BLM to the BLM-type contribution defined in the decomposition \ref{['eq:dGam2fold_BLMseperation']} at, respectively, $\mathcal{O}(\alpha_s^{2})$ and $\mathcal{O}(\alpha_s^{3})$.
• Figure 5: The scale dependence of the fixed-order results for $\mathrm{\Gamma}_t$ and $\bar{\Gamma}_t$ in $\mu/m_t \in [0.1, 1.3]$