We will produce combined resummed QCD results with full dependence on the final-state variables, and use them to construct approximate N3LO to be used for the estimation of perturbative uncertainties.

The motivation is to construct and combine high-energy resummation, transverse-momentum resummation, and threshold resummation, for the main inclusive observables used for PDF determinations, such as triple-differential rapidity, transverse-momentum, and invariant-mass distributions for gauge boson production. The truncation of these results at finite order allows then one to construct approximate finite-order cross-sections at the next unknown finite order: currently N3LO. These can be used for a reliable estimate of the missing higher order uncertainty on current results.

The main challenge is to combine consistently transverse momentum, threshold and high-energy resummation for differential distributions in such a way that upon integration over any kinematic variable consistent resummed results are obtained.

As an intermediate goal, we will construct approximate N3LO results for all processes used in PDF determination from known resummed results.

The final deliverables will include both analytic expressions for combined resummed results and approximate N3LO predictions to be used for the estimation of missing higher order uncertainties.