Analytic third-order QCD corrections to top-quark and semileptonic $b\to u$ decays

We present the first analytic results of next-to-next-to-next-to-leading-order (${\mathrm{N}}^3\mathrm{LO}$) QCD corrections to the top-quark decay width. We focus on the dominant leading color contribution, which includes light-quark loops. At next-to-next-to-leading order (NNLO), this dominant contribution accounts for 95% of the total correction. By utilizing the optical theorem, the ${\mathrm{N}}^3\mathrm{LO}$ corrections are related to the imaginary parts of the four-loop self-energy Feynman diagrams, which are calculated with differential equations. The results are expressed in terms of harmonic polylogarithms, enabling fast and accurate evaluation. The third-order QCD corrections decrease the leading-order decay width by 0.667%, and the scale uncertainty is reduced by half compared to the NNLO result. The most precise prediction for the top-quark width is now 1.321 GeV for $m_t=172.69\mathrm{GeV}$. Additionally, we obtain the third-order QCD corrections to the dilepton invariant mass spectrum and decay width in the semileptonic $b\to u$ transition. The perturbative series in the on-shell mass scheme exhibits poor convergence behavior. In the $\overline{\mathrm{MS}}$ mass scheme, the scale dependence is greatly improved. A more precise determination of the CKM matrix element $V_{ub}$ could be obtained with such higher-order corrections.

Figure 1

Top-quark width at different values of $m_t$. The black line shows the LO result. The NLO, NNLO, and ${\mathrm{N}}^3\mathrm{LO}$ predictions with QCD scale uncertainties are represented by the blue, red, and green bands, respectively.

Figure 2

The $q^2/m_b^2$ distribution for $b\to X_{u}e{\overline{\nu }}_e$ in the $\overline{\mathrm{MS}}$ mass scheme normalized by ${\mathrm{\Gamma }}_b^{(0)}$. The bands denote the scale uncertainties.

Figure 3

Scale dependence of the $b$-quark semileptonic decay width normalized by the LO width in the on-shell mass scheme. The solid and dashed lines represent the results in the $\overline{\mathrm{MS}}$ and on-shell mass schemes, respectively.