This paper reviews precise theoretical calculations for two kinds of processes at the Large Hadron Collider (LHC): the scattering of two electroweak gauge bosons (vector‑boson scattering, VBS) and the production of three gauge bosons (triboson or VVV production). The authors collect results for next‑to‑leading‑order (NLO) corrections from both the strong force (quantum chromodynamics, QCD) and the electroweak force. A central finding is that purely electroweak corrections can be large and negative: about −16% for like‑sign W±W± scattering and about −7% for triple‑W production, even for overall cross sections.

The review covers full ‘‘off‑shell’’ NLO calculations and practical approximations. Off‑shell means the intermediate W and Z bosons are treated with their real propagation and decays, so the calculations include the actual final‑state leptons and jets seen in detectors. These calculations are complex: a typical VBS process already involves roughly 10^2 Feynman diagrams at leading order and 10^3–10^4 at NLO, with very challenging loop integrals. The results presented were obtained with dedicated Monte Carlo integrators (Bonsay and MoCaNLO) combined with matrix‑element and loop libraries (OpenLoops, Recola and Collier).

At a high level, the large electroweak effects come from so‑called Sudakov logarithms. These are mathematical terms that grow with the energy flowing through the process and reduce the predicted rates at high momentum transfer. In the like‑sign WW example the typical energy scale is around Q≈400 GeV, which explains the sizable −16% effect quoted. Typical NLO calculations include several orders in the expansion in the strong and electroweak coupling constants (for example O(α7), O(αsα6), and so on). Experimental selections that tag two forward jets with large invariant mass (for example Mjj>500 GeV and a rapidity gap |Δyjj|>2.5) strongly enhance the genuine electroweak VBS signal; with such cuts the electroweak part makes about 90% of the like‑sign WW rate.