Mixed NNLO QCD×electroweak corrections of $\mathcal{O}(N_f \alpha_s \alpha)$ to single-W/Z production at the LHC

First results on the radiative corrections of order $ \mathcal{O} $(N$_{f}$α$_{s}$α) are presented for the off-shell production of W or Z bosons at the LHC, where N$_{f}$ is the number of fermion flavours. These corrections comprise all diagrams at $ \mathcal{O} $(α$_{s}$α) with closed fermion loops, form a gauge-invariant part of the next-to-next-to-leading-order corrections of mixed QCD×electroweak type, and are the ones that concern the issue of mass renormalization of the W and Z resonances. The occurring irreducible two-loop diagrams, which involve only self-energy insertions, are calculated with current standard techniques, and explicit analytical results on the electroweak gauge-boson self-energies at $ \mathcal{O} $(α$_{s}$α) are given. Moreover, the generalization of the complex-mass scheme for a gauge-invariant treatment of the W/Z resonances is described for the order $ \mathcal{O} $(α$_{s}$α). While the corrections, which are implemented in the Monte Carlo program Rady, are negligible for observables that are dominated by resonant W/Z bosons, they affect invariant-mass distributions at the level of up to 2% for invariant masses of ≳ 500 GeV and are, thus, phenomenologically relevant. The impact on transverse-momentum distributions is similar, taking into account that leading-order predictions to those distributions underestimate the spectrum.