Higgs boson production at the LHC using the $q_T$ subtraction formalism at N$^3$LO QCD

We consider higher-order QCD corrections to Higgs boson production through gluon-gluon fusion in the large top quark mass limit in hadron collisions. We extend the transverse-momentum (q$_{T}$ ) subtraction method to next-to-next-to-next-to-leading order (N$^{3}$LO) and combine it with the NNLO Higgs-plus-jet calculation to numerically compute differential infrared-safe observables at N$^{3}$LO for Higgs boson production in gluon fusion. To cancel the infrared divergences, we exploit the universal behaviour of the associated q$_{T}$ distributions in the small-q$_{T}$ region. We document all the necessary ingredients of the transverse-momentum subtraction method up to N$^{3}$LO. The missing third-order collinear 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.