Combining QED and QCD transverse-momentum resummation for W and Z boson production at hadron colliders

Abstract: In this article, we consider the transverse momentum ($q_T$) distribution of $W$ and $Z$ bosons produced in hadronic collisions. We combine the $q_T$ resummation for QED and QCD radiation including the QED soft emissions from the $W$ boson in the final state. In particular, we perform the resummation of enhanced logarithmic contributions due to soft and collinear emissions at next-to-leading accuracy in QED, leading-order accuracy for mixed QED-QCD and next-to-next-to-leading accuracy in QCD. In the small-$q_T$ region we consistently include in our results the next-to-next-to-leading order (i.e.\ two loops) QCD corrections and the next-to-leading order (i.e.\ one loop) electroweak corrections. The matching with the fixed-order calculation at large $q_T$ has been performed at next-to-leading order in QCD (i.e.\ at $\mathcal{O}(\alpha_S^2)$) and at leading order in QED. We show numerical results for $W$ and $Z$ production at the Tevatron and the LHC. Finally, we consider the effect of combined QCD and QED resummation for the ratio of $W$ and $Z$ $q_T$ distributions, and we study the impact of the QED corrections providing an estimate of the corresponding perturbative uncertainties.