Abstract
We consider Drell-Yan production pp → Z/γ∗ → ℓ+ℓ− with the simultaneous measurement of the Z-boson transverse momentum qT and 0-jettiness {mathcal{T}}_0 . Since both observables resolve the initial-state QCD radiation, the double-differential cross section in qT and {mathcal{T}}_0 contains Sudakov double logarithms of both qT/Q and {mathcal{T}}_0/Q , where Q ∼ mZ is the dilepton invariant mass. We simultaneously resum the logarithms in qT and {mathcal{T}}_0 to next-to-next-to-leading logarithmic order (NNLL) matched to next-to-leading fixed order (NLO). Our results provide the first genuinely two-dimensional analytic Sudakov resummation for initial-state radiation. Integrating the resummed double-differential spectrum with an appropriate scale choice over either {mathcal{T}}_0 or qT recovers the corresponding single-differential resummation for the remaining variable. We discuss in detail the required effective field theory setups and their combination using two-dimensional resummation profile scales. We also introduce a new method to perform the qT resummation where the underlying resummation is carried out in impact-parameter space, but is consistently turned off depending on the momentum-space target value for qT. Our methods apply at any order and for any color-singlet production process, such that our results can be systematically extended when the relevant perturbative ingredients become available.
Highlights
Widely separated, the perturbative series at each order is dominated by logarithms of their ratios
An important feature of our effective field theories (EFTs) setup is that the factorized cross section in soft-collinear effective theory (SCET)+ differs from the ones in SCETI and SCETII only by a subset of the power corrections it receives relative to the full QCD result, dσI
We demonstrate that our prediction smoothly interpolates between the SCETI and SCETII boundary theories, i.e., we show that our matching formula in eq (3.5) recovers the matched predictions on either boundary and improves over them by an additional resummation of power-suppressed terms
Summary
We consider color-singlet production at hadron colliders. the process dependence is not important for our discussion, we consider the example of Drell-Yan production, pp → Z/γ∗(→ + −), for concreteness. Approaching qT ∼ T ∼ Q, the qT resummation must again be turned off to ensure the delicate cancellations between singular and nonsingular contributions and to properly recover the correct fixed-order result for the spectrum We achieve this by constructing hybrid profile scales that depend on both bT and qT , and undergo a continuous deformation away from the canonical bT scales in eq (2.24) as a function of the target qT value, schematically, μIBI,S(qT , bT ) , νBII,S(qT , bT ) → μIHI = μFO for qT → Q. As for SCETI, ∆FO is estimated by overall variations of μFO by a factor of two, which is inherited by all SCETII scales, so it probes the fixed-order uncertainties while leaving the resummed logarithms invariant. Our framework to match between the rich logarithmic structure predicted by eq (2.39) and the two boundary regimes is the subject of section 3
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