Momentum-space resummation for transverse observables and the Higgs $p_\perp$ at N$^3$LL+NNLO

We present an approach to the momentum-space resummation of global, recursive infrared and collinear safe observables featuring kinematic zeros away from the Sudakov limit. In the hadro-production of a generic colour singlet, we consider the family of inclusive observables which do not depend on the rapidity of the radiation, prime examples being the transverse momentum of the singlet, and $\phi^*$ in Drell-Yan pair production. We derive a resummation formula valid up to next-to-next-to-next-to-leading-logaritmic accuracy for the considered observables. This formula reduces exactly to the customary resummation performed in impact-parameter space in the known cases, and it also predicts the correct power-behaved scaling of the cross section in the limit of small value of the observable. We show how this formalism is efficiently implemented by means of Monte Carlo techniques in a fully exclusive generator that allows one to apply arbitrary cuts on the Born variables for any colour singlet, as well as to automatically match the resummed results to fixed-order calculations. As a phenomenological application, we present state-of-the-art predictions for the Higgs-boson transverse-momentum spectrum at the LHC at next-to-next-to-next-to-leading-logarithmic accuracy matched to fixed next-to-next-to-leading order.