The Two-Loop Radiative Gluon Jet Function for $gg\to h$ via a Light Quark Loop

The Higgs-boson production channel $gg\to h$ mediated by light-quark loops receives large logarithmic corrections in QCD, which can be resummed using factorization formulae derived in soft-collinear effective theory. In these factorization formulae the radiative gluon jet function appears, which is a central object in the study of factorization beyond the leading order in scale ratios. We calculate this function at two-loop order for the first time and present the subtleties that come along with this.