Two-loop electroweak corrections to Higgs–gluon couplings to higher orders in the dimensional regularization parameter

We compute the two-loop electroweak correction to the production of the Higgs boson in gluon fusion to higher orders in the dimensional-regularization parameter $\epsilon =(d-4)/2$. We employ the method of differential equations augmented by the choice of a canonical basis to compute the relevant integrals and express them in terms of Goncharov polylogarithms. Our calculation provides useful results for the computation of the NLO mixed QCD-electroweak corrections to $gg\to H$ and establishes the necessary framework towards the calculation of the missing three-loop virtual corrections.