Higgs boson decay to massive bottom quarks at order ${\alpha }_s^4$ induced by top-quark Yukawa couplings

The Higgs boson decay to massive bottom quarks has the largest branching ratio. The decay is mainly induced by the bottom-quark Yukawa coupling with the decay rate calculated up to $\mathcal{O}({\alpha }_s^4)$ assuming the massless final-state bottom quark. The top-quark Yukawa coupling induced contribution starts at $\mathcal{O}({\alpha }_s^2)$, and exhibits logarithmic and power enhancements, making the perturbative expansion converge slowly. We present a calculation of such contributions at $\mathcal{O}({\alpha }_s^4)$ to the decay into massive bottom quarks in which the squared amplitudes contain two top-quark Yukawa couplings. We find that they increase the decay width, relative to the result up to $\mathcal{O}({\alpha }_s^3)$, by $0.4\mathrm{\%}$, larger than the experimental precision at future lepton colliders, and reduce the scale dependence significantly down to $0.4\mathrm{\%}$.