Joint Two-Dimensional Resummation in $q_T$ and $0$-Jettiness at NNLL

Abstract

We consider Drell-Yan production $pp\to Z/{\gamma }^{\ast }\to {\ell }^{+}{\ell }^{-}$ with the simultaneous measurement of the $Z$-boson transverse momentum $q_T$ and $0$-jettiness ${\mathcal{T}}_0$. Since both observables resolve the initial-state QCD radiation, the double-differential cross section in $q_T$ and ${\mathcal{T}}_0$ contains Sudakov double logarithms of both $q_T/Q$ and ${\mathcal{T}}_0/Q$, where $Q\sim m_Z$ is the dilepton invariant mass. We simultaneously resum the logarithms in $q_T$ 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 $q_T$ 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.

Comments: 43 pages + 10 pages in appendices, 20 figures; v2: journal version; references added, panel with fixed-order result added to figure 15; v3: fixed a typo (incorrect sign) in eq. (B.19)