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The $H\to gg$ amplitude relevant for Higgs production via gluon fusion iscomputed in the four-dimensional helicity scheme (FDH) and in dimensionalreduction (DRED) at the two-loop level. The required renormalization isdeveloped and described in detail, including the treatment of evanescent$\epsilon$-scalar contributions. In FDH and DRED there are additionaldimension-5 operators generating the $H g g$ vertices, where $g$ can either bea gluon or an $\epsilon$-scalar. An appropriate operator basis is given and theoperator mixing through renormalization is described. The results of thepresent paper provide building blocks for further computations, and they allowto complete the study of the infrared divergence structure of two-loopamplitudes in FDH and DRED.

Computation of $H\to gg$ in FDH and DRED: renormalization, operator mixing, and explicit two-loop results