The transverse-momentum subtraction method at N$^3$LO applied to Higgs boson production at the LHC

We consider the extension of the transverse-momentum ($q_T$) subtraction method at next-to-next-to-next-to-leading order (N$^3$LO) in perturbative QCD. While all the $q_T$-subtraction ingredients at $q_T \neq 0$ are known in analytical form, the third-order collinear functions and helicity-flip functions, which contribute only at $q_T=0$, are approximated using a prescription which uses the known result for the total Higgs boson cross section at this order. As a first application of the third-order $q_T$-subtraction method, we present the N$^3$LO rapidity distribution of the Higgs boson at the LHC.