The production of weak gauge bosons, W± 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 qT of the lepton pair and its azimuthal separation ∆ϕ. 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 qT and ∆ϕ as well as the double-differential (qT, ∆ϕ) spectra for all three processes at N3LL′+N2LO 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 ℛW+/Z\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathrm{\mathcal{R}}}_{W^{+}/Z} $$\end{document}, ℛW−/Z\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathrm{\mathcal{R}}}_{W^{-}/Z} $$\end{document}, and ℛW+/W−\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathrm{\mathcal{R}}}_{W^{+}/{W}^{-}} $$\end{document} 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 qT region.
