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 $\mathrm{p}\mathrm{p}\to \mathrm{\ell }{\nu }_{\mathrm{\ell }}+4\mathrm{j}$ in two fiducial regions which are particularly relevant for its experimental measurement. In addition to the fully electroweak order, i.e. $\mathcal{O}({\alpha }^6)$, we complement our results with $\mathcal{O}({\alpha }_{\mathrm{s}}{\alpha }^5)$ and $\mathcal{O}({\alpha }_{\mathrm{s}}^2{\alpha }^4)$ for inclusive predictions. At $\mathcal{O}({\alpha }^6)$ 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.