Combined NLO EW and QCD corrections to off-shell $t\overline{t}$W production at the LHC

The high luminosity that will be accumulated at the LHC will enable precise differential measurements of the hadronic production of a top-antitop-quark pair in association with a W boson. Therefore, an accurate description of this process is needed for realistic final states. In this work we combine for the first time the NLO QCD and electroweak corrections to the full off-shell $t\overline{t}$W${}^{+}$ production at the LHC in the three-charged-lepton channel, including all spin correlations, non-resonant effects, and interferences. To this end, we have computed the NLO electroweak radiative corrections to the leading QCD order as well as the NLO QCD corrections to both the QCD and the electroweak leading orders.