The study of top-quark properties will be a central aspect of the physics programme of any future lepton collider. In this article, we investigate the production of top-quark pairs in the semi-leptonic decay channel in e+e-\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\textrm{e}}^{+}\textrm{e}^{-}$$\end{document} collisions, whose experimental signature is one charged lepton, jets, and missing energy. We present for the first time fiducial cross sections and differential distributions at next-to-leading-order accuracy in QCD for the full off-shell process. We find that the QCD corrections for the considered process are strongly dependent on the beam energies and range from few per cent up to more than 100%\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$100\%$$\end{document} (near threshold and above 1TeV\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$1\,\text {TeV} $$\end{document}). We focus, in particular, on two scenarios: one close to threshold (365GeV\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$365\,\text {GeV} $$\end{document}), dominated by top-pair production, and one at the TeV scale (1.5TeV\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$1.5\,\text {TeV} $$\end{document}), for which irreducible-background contributions become relevant. An assessment of polarised-beam effects is also provided.
