Abstract
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 {mathrm{mathcal{R}}}_{W^{+}/Z} , {mathrm{mathcal{R}}}_{W^{-}/Z} , and {mathrm{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 qT region.
Highlights
The measurement of W production always involves the determination of the event’s missing momentum as a proxy for the inaccessible neutrino momentum
As the renormalisability of SM relies on the anomaly cancellation within each quark generation, the removal of the top quark by brute force will break the renormalisation group invariance (RGI) explicitly and it necessitates an additional renormalisation constant in the low energy effective field theory (LEEFT) for its restoration
The uncertainty of the cross section ratios follows the pattern laid out in their definition: while the fully-correlated case leads to vanishingly small uncertainties, smaller than 1% in most regions, the fully-uncorrelated case lies on the opposite end of the spectrum with uncertainties of ±4% for our best calculation at N3LL +N2LO accuracy both in the fixed-order region and the resummation region
Summary
From the QCD factorisation theorem [127], the differential cross section for the Drell-Yan (DY) process can be expressed as d5σ d2qT dYL dML2 dΩL. Σij denotes the partonic cross section which depends on the renormalisation scale μR as well as the factorisation scale μF . In light of its particular performance in the fixed-order calculations, the framework with the exponential regulator will be employed in this work. In this approach, the factorisation formula of eq (2.1) can be rewritten as [101, 102], 16s(2π)4ML2 i,j d2bT eibT·qT HiVj (ML, ΩL, μR) S(bT, μR, ν). In addition to the soft and beam functions, the factorisation formula in eq (2.4) involves the hard sector HiVj (V = γ/Z, W ±).
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