Analytic result for the top-quark width at next-to-next-to-leading order in QCD

Abstract

We present the first full analytic results of next-to-next-to-leading order (NNLO) QCD corrections to the top-quark decay width $\mathrm{\ensuremath{\Gamma}}(t\ensuremath{\rightarrow}Wb)$ by calculating the imaginary part of three-loop top-quark self-energy diagrams. The results are all expressed in terms of harmonic polylogarithms and valid in the whole region $0\ensuremath{\le}{m}_{W}^{2}\ensuremath{\le}{m}_{t}^{2}$. The expansions in the ${m}_{W}^{2}\ensuremath{\rightarrow}0$ and ${m}_{W}^{2}\ensuremath{\rightarrow}{m}_{t}^{2}$ limits coincide with previous studies. Our results can also be taken as the exact prediction for the lepton invariant mass spectrum in semileptonic $b\ensuremath{\rightarrow}u$ decays. We also analytically compute the decay width including the off shell $W$ boson effect up to NNLO in QCD for the first time. Combining these contributions with electroweak corrections and the finite $b$-quark mass effect, we determine the most precise top-quark width to be 1.331 GeV for ${m}_{t}=172.69\text{ }\text{ }\mathrm{GeV}$. The total theoretical uncertainties including those from renormalization scale choice, top-quark mass renormalization scheme, input parameters, and missing higher-order corrections are scrutinized and found to be less than 1%.