Probing top-quark couplings indirectly at Higgs factories

We perform a global effective-field-theory analysis to assess the combined precision on Higgs couplings, triple gauge-boson couplings, and top-quark couplings, at future circular e${}^{+}$e${}^{-}$ colliders, with a focus on runs below the t$\overline{t}$ production threshold. Deviations in the top-quark sector entering as one-loop corrections are consistently taken into account in Higgs and diboson processes. We find that future lepton colliders running at center-of-mass energies below the t$\overline{t}$ production threshold can still provide useful information on top-quark couplings, by measuring virtual top-quark effects. With rate and differential measurements, the indirect individual sensitivity achievable is better than at the high-luminosity LHC. However, strong correlations between the extracted top-quark and Higgs couplings are also present and lead to much weaker global constraints on top-quark couplings. This implies that a direct probe of top-quark couplings above the t$\overline{t}$ production threshold is helpful also for the determination of Higgs and triple-gauge-boson couplings. In addition, we find that below the e${}^{+}$e${}^{-}$ $\to$ t$\overline{t}$h production threshold, the top-quark Yukawa coupling can be determined by its loop corrections to all Higgs production and decay channels. Degeneracy with the ggh coupling can be resolved, and even a global limit is competitive with the prospects of a linear collider above the threshold. This provides an additional means of determining the top-quark Yukawa coupling indirectly at lepton colliders.