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 ($\mathcal{O}(α_s^3)$) 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 $B \to X_u \ell ν$ width, crucial for a percent-level determination of $|V_{ub}|$, and the first $\mathcal{O}(α_s^3)$ 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.

Figures (3)

• Figure 1: Top row: from left to right are the plots for the numerical results of $W_1\,, W_2$ in a regular phase-space region $R_2$ up to $\mathcal{O}(\alpha_s^{3})$. Middle row: from left to right are for $W_3\,, W_4$. Bottom row: from left to right are for $W_5$ and the distribution \ref{['eq:2Dparametric']} plotted in the similar way.
• Figure 2: The left plot shows the perturbative coefficients after subtracting $\mathcal{O}(\alpha_s^{N+1} \beta_0^{N})$ terms with $N \geq 1$, with the $\mathcal{O}(\alpha_s^{1})$-coefficient included for reference. The right plot shows the ratios of the non-BLM to the BLM-type contribution at, respectively, $\mathcal{O}(\alpha_s^{2})$ and $\mathcal{O}(\alpha_s^{3})$.
• Figure 3: The perturbative results for the histogrammed $q^2$-spectrum determined in the kinetic mass scheme at different orders in $\alpha_s$ with the conventional scale variation $\mu \in [m_{b,\mathrm{kin}}/2, m_{b,\mathrm{kin}}]$.