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We present a two-loop analysis of the contributions to Higgs production via gluon-gluon fusion arising from the triple-gluon operator in the Standard Model effective field theory (SMEFT). Our discussion covers all aspects of renormalization group (RG) improved perturbation theory, including matching and running within the SMEFT. This study can therefore be seen as a blueprint of the intricacies and subtleties that arise in RG improved SMEFT calculations for collider processes beyond the leading order.

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