One-loop corrections to $h\to b\bar b$ and $h\to \tau\bar \tau$ decays in the Standard Model Dimension-6 EFT: four-fermion operators and the large-$m_t$ limit

We calculate a set of one-loop corrections to $h\to b\bar b$ and $h\to \tau\bar \tau$ decays in the dimension-6 Standard Model effective field theory (SMEFT). In particular, working in the limit of vanishing gauge couplings, we calculate directly in the broken phase of the theory all large logarithmic corrections and in addition the finite corrections in the large-$m_t$ limit. Moreover, we give exact results for one-loop contributions from four-fermion operators. We obtain these corrections within an extension of the widely used on-shell renormalisation scheme appropriate for SMEFT calculations, and show explicitly how UV divergent bare amplitudes from a total of 21 different SMEFT operators are rendered finite within this scheme. As a by-product of the calculation, we also compute to one-loop order the logarithmically enhanced and finite large-$m_t$ corrections to muon decay in the limit of vanishing gauge couplings, which is necessary to implement the $G_F$ input parameter scheme within the SMEFT.