The $q_T$ and $\Delta\phi$ spectra in W and Z production at the LHC at N$^3$LL'+N$^2$LO

The production of weak gauge bosons, $W^{\pm}$ and $Z$, are at the core of the LHC precision measurement program. Their transverse momentum spectra as well as their pairwise ratios are key theoretical inputs to many high-precision analyses, ranging from the $W$ mass measurement to the determination of parton distribution functions. Owing to the different properties of the $W$ and $Z$ boson and the different accessible fiducial regions for their measurement, a simple one-dimensional correlation is insufficient to capture the differing vector and axial-vector dynamics of the produced lepton pair. We propose to correlate them in two observables, the transverse momentum $q_T$ of the lepton pair and its azimuthal separation $\Delta\phi$. Both quantities are purely transverse and therefore accessible in all three processes, either directly or by utilising the missing transverse momentum of the event. We calculate all the single-differential $q_T$ and $\Delta\phi$ as well as the double-differential $(q_T,\Delta\phi)$ spectra for all three processes at N$^3$LL'+N$^2$LO accuracy, resumming small transverse momentum logarithms in the soft-collinear effective theory approach and including all singlet and non-singlet contributions. Using the double-differential cross sections we build the pairwise ratios $\mathcal{R}_{W^+/Z},~\mathcal{R}_{W^-/Z},~\mathcal{R}_{W^+/W^-}$ and determine their uncertainties assuming fully correlated, partially correlated, and uncorrelated uncertainties in the respective numerators and denominators. In the preferred partially correlated case we find uncertainties of less than 1% in most phase space regions and up to 3% in the lowest $q_T$ region.