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The process ${\rm p}{\rm p}\to\mu^+\nu_\mu{\rm e}^+\nu_{\rm e}{\rm j}{\rm j}$receives several contributions of different orders in the strong andelectroweak coupling constants. Using appropriate event selections, thisprocess is dominated by vector-boson scattering (VBS) and has recently beenmeasured at the LHC. It is thus of prime importance to estimate precisely eachcontribution. In this article we compute for the first time the full NLO QCDand electroweak corrections to VBS and its irreducible background processeswith realistic experimental cuts. We do not rely on approximations but usecomplete amplitudes involving two different orders at tree level and threedifferent orders at one-loop level. Since we take into account allinterferences, at NLO level the corrections to the VBS process and to theQCD-induced irreducible background process contribute at the same orders. Hencethe two processes cannot be unambiguously distinguished, and all contributionsto the $\mu^+\nu_\mu{\rm e}^+\nu_{\rm e}{\rm j}{\rm j}$ final state should bepreferably measured together.

Complete NLO corrections to ${\rm W}^+{\rm W}^+$ scattering and its irreducible background at the LHC