The parton-level structure of Higgs decays to hadrons at N$^3$LO

We present the quantum chromodynamics (QCD) corrections for Higgs boson decays to hadronic final states at next-to-next-to-next-to-leading order (N$^3$LO) in the strong coupling constant $\alpha_s$. In particular, we consider the Higgs boson decay to massless bottom quarks and the Higgs boson decay to a pair of gluons in the limit of a heavy top quark. The tree-level five-parton, the one-loop four-parton, the two-loop three-parton, and the three-loop two-parton matrix elements are integrated separately over the inclusive phase space and classified by partons appearing in the final state and by colour structure. As a check, we reproduce known results for the hadronic $R$-ratios at N$^3$LO. We study patterns of infrared singularity cancellation within the colour layers of the integrated expressions and observe an agreement in the highest trascendental weight terms in the decay of different colour singlets to quarks. We anticipate that our result will be an essential ingredient for the formulation of N$^3$LO subtraction schemes.