QCD radiative corrections for $h\to b\bar b$ in the Standard Model Dimension-6 EFT

We calculate the $\mathcal{O}(\alpha_s)$ QCD corrections to the inclusive $h\to b\bar b$ decay rate in the dimension-6 Standard Model Effective Field Theory (SMEFT). The QCD corrections multiplying the dimension-6 Wilson coefficients which alter the $hb\bar b$-vertex at tree-level are proportional to the Standard Model (SM) ones, so next-to-leading order results can be obtained through a simple rescaling of the tree-level decay rate. On the other hand, contributions from the operators $Q_{bG}$ and $Q_{HG}$, which alter the $gb\bar b$-vertex and introduce a $hgg$-vertex respectively, enter at $\mathcal{O}(\alpha_s)$ and induce sizeable corrections which are unrelated to the SM ones and cannot be anticipated through a renormalisation-group analysis. We present compact analytic results for these contributions, which we recommend to be included in future phenomenological studies.