Precision tests of third-generation four-quark operators: $gg\to h$ and $h\to \gamma \gamma$

We compute the two-loop contributions to Higgs production via gluon-gluon fusion ($gg\to h$) and Higgs decay into two photons ($h\to \gamma \gamma$), arising from third-generation four-quark operators in the Standard Model effective field theory (SMEFT). Our analysis is performed in the broken phase of the theory, retaining the full dependence on the Higgs and heavy-quark masses. This includes both finite matching corrections and logarithmic effects stemming from the renormalization group evolution within the SMEFT. As a byproduct, new two-loop anomalous dimensions in the SMEFT are obtained. We also briefly discuss the phenomenological implications of our two-loop calculations.