A detailed study on the prospects for a $\mathrm{t\overline{t}}$ threshold scan in $\mathrm{e^+e^-}$ collisions

A scan of the beam energy across the top quark pair ($\mathrm{t\overline{t}}$) production threshold is part of the program of future Higgs, top, and electroweak factory projects. In this paper, we provide projections for the achievable precision in the top quark mass ($m_\mathrm{t}$), width ($\Gamma_\mathrm{t}$), and Yukawa coupling ($y_\mathrm{t}$) at the electron-positron ($\mathrm{e^+e^-}$) stage of the Future Circular Collider (FCC-ee). The study includes a detailed assessment of parametric and systematic uncertainties, as well as a rigorous estimate of the effect of point-to-point correlations. We project that $m_\mathrm{t}$ and $\Gamma_\mathrm{t}$ can be determined with an experimental precision of about 6.8 and 11.5 MeV, respectively, when $m_\mathrm{t}$ is defined in the potential-subtracted (PS) scheme. The impact of theoretical uncertainties due to missing higher orders is found to be of about 35 (25) MeV on $m_\mathrm{t}$ ($\Gamma_\mathrm{t}$) at N$^3$LO in non-relativistic QCD. Therefore, improvements in the theoretical accuracy, which is an active area of development, are key to match the achievable experimental precision at a future $\mathrm{e^+e^-}$ collider. Finally, we explore the prospects for a measurement of $y_\mathrm{t}$ at FCC-ee via a dedicated run above the $\mathrm{t\overline{t}}$ production threshold.