I show how to construct an approximate expression for the cross section for Higgs production in gluon fusion at NNNLO with finite top mass, exploiting existing knowledge on its singularity structure in Mellin space. I show how the dominant and some subdominant singularities can be extracted from Sudakov and BFKL resummation, and I use them to construct approximate results up to NNNLO. By comparing to known exact NLO and NNLO results I show that this method works quite well, and specifically better than other existing approximations.
I present a prediction for the total cross section, and show that the NNNLO leads to a correction to the NNLO result of order 15%, which substantially reduces its scale dependence.
