Abstract
We present the complete next-to-leading order (NLO) QCD and NLO electroweak (EW) corrections to the $W$-boson pair production associated with a hard jet at the LHC including the spin correlated $W$-boson leptonic decays. The dependence of the leading order (LO) and NLO corrected cross sections on the jet transverse momentum cut is investigated. In dealing with the subsequent resonant $W$-boson decays, we adopt the {\sc MadSpin} method where the spin correlation effect is included. We also provide the LO and NLO corrected distributions of the transverse momenta and rapidities of final products. The results show that the NLO EW correction is significant in high transverse momentum region due to the EW Sudakov effect.
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