An approximate NNNLO cross section for Higgs production in gluon fusion

An approximate expression for the inclusive Higgs production cross section in gluon fusion at NNNLO in QCD with finite top mass is presented. We argue that an accurate approximation can be constructed combining (and improving) the large- and small-z behaviours of the partonic cross section, which are both known to all orders from soft-gluon (Sudakov) and high-energy (BFKL) resummations, respectively. For a 125 GeV Higgs at LHC at 8 TeV, we find an increase of about 6-13% with respect to the NNLO inclusive cross section for the conventional scale \mu_R=m_H/2, suggesting that higher order QCD corrections might be underestimated by presently available results. We also find a significant reduction of the scale uncertainty.