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

Abstract

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 $\Delta \varphi$. 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 $\Delta \varphi$ as well as the double-differential $(q_T,\Delta \varphi )$ 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.

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