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

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 and electroweak coupling constants. Using appropriate event selections, this process is dominated by vector-boson scattering (VBS) and has recently been measured at the LHC. It is thus of prime importance to estimate precisely each contribution. In this article we compute for the first time the full NLO QCD and electroweak corrections to VBS and its irreducible background processes with realistic experimental cuts. We do not rely on approximations but use complete amplitudes involving two different orders at tree level and three different orders at one-loop level. Since we take into account all interferences, at NLO level the corrections to the VBS process and to the QCD-induced irreducible background process contribute at the same orders. Hence the two processes cannot be unambiguously distinguished, and all contributions to the $\mu^+\nu_\mu{\rm e}^+\nu_{\rm e}{\rm j}{\rm j}$ final state should be preferably measured together.