The \(\msbar\) Renormalized Bottom Mass of Order \({\cal O}(\alpha_s G_F M_t^2)\) and its Application to \(\Gamma(H\to b\bar{b})\)

The renormalized mass of the bottom quark is calculated at the two loop level to order \({\cal O}(\alpha_s G_F M_t^2)\) in the \(\msbar\) renormalization scheme. Different strategies for the computation are outlined. The result is applied to the partial decay rate \(\Gamma(H\rightarrow b\bar{b})\) of the Higgs boson into bottom quarks. Expressing the width in terms of the running mass instead of the bottom pole mass allows to treat the \({\cal O}(\alpha_s G_F M_t^2)\) radiative corrections on the same footing as is commonly used in pure QCD calculations. The numerical values for the corrections are given and the sizes of different contributions are compared.