Prospects for the determination of the top-quark Yukawa coupling at future $e^+e^-$ colliders

We estimate the sensitivity to the top-quark Yukawa coupling $y_t$ at future $e^+e^-$ colliders. We go beyond the standard approach that focuses on $t\bar t h$ production and consider final states with a Higgs boson but not top quarks. The sensitivity to $y_t$ in such processes comes from the coupling of the Higgs boson to top quarks in loops. Such final states can be produced in significant numbers at center-of-mass energies that will be accessible by all proposed $e^+e^-$ colliders. In a simplified theoretical framework to parametrise deviations from the Standard Model, we find that at FCC-$ee$ and CEPC operating at $\sqrt{s}=240$ GeV, $y_t$ could potentially be measured with precision better than $1\%$. For CLIC and ILC the extraction of $y_t$ could be improved by a factor of about 2 and 7 respectively, compared to its extraction from just $t\bar t h$ final states.