Double-pole approximation for leading-order semi-leptonic vector-boson scattering at the LHC

Measuring vector-boson scattering beyond the fully-leptonic final state is becoming possible at the LHC, which demands to have a solid control on the theory predictions for all final states of this class of processes. In this work we present a full off-shell leading-order calculation for the process pp → ℓ⁺ν_ℓ + 4j in two fiducial regions which are particularly relevant for its experimental measurement. In addition to the fully electroweak order, i.e. $\mathcal{O}\left({\alpha }^6\right)$ \( \mathcal{O}\left({\alpha}^6\right) \) , we complement our results with $\mathcal{O}\left({\alpha }_{\mathrm{s}}{\alpha }^5\right)$ \( \mathcal{O}\left({\alpha}_{\textrm{s}}{\alpha}^5\right) \) and $\mathcal{O}\left({\alpha }_{\mathrm{s}}^2{\alpha }^4\right)$ \( \mathcal{O}\left({\alpha}_{\textrm{s}}^2{\alpha}^4\right) \) for inclusive predictions. At $\mathcal{O}\left({\alpha }^6\right)$ \( \mathcal{O}\left({\alpha}^6\right) \) we present for the first time a systematic treatment of the process in double-pole approximation and we perform a detailed study of its range of validity by considering inclusive and differential predictions compared to the full off-shell calculation.