Probability-based Estimates of the Uncalculated N${}^5$LO QCD Contribution to the Hadronic $W$-Boson Decay Width

Abstract

The perturbative QCD corrections to the hadronic decay width of the $W$ boson are currently known up to next-to-next-to-next-to-next-to-leading order (${\mathrm{N}}^4\mathrm{LO}$), whereas the exact ${\mathrm{N}}^5\mathrm{LO}$ correction remains unavailable owing to its formidable computational complexity. In this work, we estimate the ${\mathrm{N}}^5\mathrm{LO}$ contributions by adopting Bayesian analysis (BA). Before performing the estimation, the Principle of Maximum Conformality (PMC) is employed to improve the precision of the initial scale-dependent perturbative series. Through recursive application of the renormalization group equation, non-conformal terms are absorbed into the strong running coupling, yielding a scheme-independent, scale-invariant perturbative series with improved convergence. The PMC procedure determines an effective coupling ${\alpha }_s(Q_{\ast })$, with the PMC scale fixed as $Q_{\ast }=100.102\mathrm{GeV}$ at next-to-next-to-leading logarithmic accuracy. Based on the improved and more precise series, the $95.5\%$ BA credible interval yields an uncertainty of $\Delta {\delta }_{\mathrm{QCD}}{|}_{\mathrm{PMC},\mathrm{BA}}^{{\mathrm{N}}^5\mathrm{LO}}=\pm 1.0\times 10^{-5}$. The resulting hadronic branching ratio is $\mathcal{B}(W\to \text{hadrons}){|}_{\mathrm{PMC}}=(65.84\pm 1.54)\%$, which is consistent with experimental data within reasonable errors.

Comments: 11 pages, 9 figures