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$Gamma(H\to b\bar{b})$ to order $αα_s$

We compute the decay rate of the Standard Model Higgs boson to bottom quarks to order $αα_s$. We apply the optical theorem and calculate the imaginary part of three-loop corrections to the Higgs boson propagator using asymptotic expansions in appropriately chosen mass ratios. The corrections of order $αα_s$ are of the same order of magnitude as the ${\cal O}(α_s^3)$ QCD corrections but have the opposite sign.