Joint two-dimensional resummation in qT and 0-jettiness at NNLL

We consider Drell-Yan production pp → Z/γ ^∗ → ℓ ⁺ ℓ ⁻ with the simultaneous measurement of the Z-boson transverse momentum q _T and 0-jettiness $$ {\mathcal{T}}_0 $$ T 0 . Since both observables resolve the initial-state QCD radiation, the double-differential cross section in q _T and $$ {\mathcal{T}}_0 $$ T 0 contains Sudakov double logarithms of both q _T /Q and $$ {\mathcal{T}}_0/Q $$ T 0 / Q , where Q ∼ m _Z is the dilepton invariant mass. We simultaneously resum the logarithms in q _T and $$ {\mathcal{T}}_0 $$ 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 $$ {\mathcal{T}}_0 $$ T 0 or q _T 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 q _T 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 q _T . 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.