Higher order corrections to mixed QCD-EW contributions to Higgs production in gluon fusion

We present an estimate of the next-to-leading order QCD corrections to mixed QCD-electroweak contribution to Higgs boson production cross section in gluon fusion, combining the recently computed three-loop virtual corrections and the approximate treatment of real emission in the soft approximation. We find that the NLO QCD corrections to mixed QCD-electroweak contributions are nearly identical to NLO QCD corrections to QCD Higgs production. Our result confirms an earlier estimate of these $\mathcal{O}\left(\alpha {\alpha }_s^2\right)$ effects in arXiv:0811.3458 [hep-ph] and provides further support for the factorization approximation of QCD and electroweak corrections.