Drell-Yan lepton-pair production: $q_T$ resummation at approximate N${}^4$LL+N${}^4$LO accuracy

We consider Drell-Yan lepton pairs produced in hadronic collisions. We present high-accuracy QCD predictions for the transverse-momentum ($q_T$) distribution and fiducial cross sections in the small $q_T$ region. We resum to all perturbative orders the logarithmically enhanced contributions up to the next-to-next-to-next-to-next-to-leading logarithmic (N${}^4$LL) accuracy and we include the hard-virtual coefficient at the next-to-next-to-next-to-leading order (N${}^3$LO) (i.e. $\mathcal{O}({\alpha }_S^3)$) with an approximation of the N${}^4$LO coefficients. The massive axial-vector and vector contributions up to three loops have also been consistently included. The resummed partonic cross section is convoluted with approximate N${}^3$LO parton distribution functions. We show numerical results at LHC energies of resummed $q_T$ distributions for $Z\mathrm{/}{\gamma }^{\ast },W^{\pm }$ production and decay, including the $W^{\pm }$ and $Z\mathrm{/}{\gamma }^{\ast }$ ratio, estimating the corresponding uncertainties from missing higher orders corrections and from incomplete or missing perturbative information coefficients at N${}^4$LL and N${}^4$LO. Our resummed calculation has been encoded in the public numerical program DYTurbo.