N$^3$LO Antenna Functions for Final-State Radiation

We discuss recent progress in the calculation of integrated antenna functions for final-state radiation at N$^3$LO in QCD. Antenna functions are directly extracted from physical matrix elements for the decay of a colour-singlet state into partons. In order to compute their integrated counterparts, which are necessary ingredients for fixed-order calculations in QCD, each allowed final state has to be individually integrated over the inclusive phase space of the unresolved radiation. We analytically integrated two-, three-, four- and five-particle final-state matrix elements for the decay of a virtual photon and a Higgs boson into partons, considering up to third-order corrections in the strong coupling constant $\alpha_s$. The integration over the inclusive phase space is performed relying on well-established reverse unitarity and IBP reduction techniques. We conclude with preliminary comments about the structure of N$^3$LO subtraction terms in the context of the antenna subtraction method.