We present the first calculation of the full next-to-leading-order electroweak and QCD corrections for vector-boson scattering (VBS) into a pair of Z bosons at the LHC. We consider specifically the process ${rm ppto e^{+}e^{-}mu^{+}mu^{-}jj}+X$ at orders $mathcal{O}(alpha^7)$ and $mathcal{O}(alpha_salpha^6)$ and take all off-shell and interference contributions into account. Owing to the presence of enhanced Sudakov logarithms, the electroweak corrections amount to $-16%$ of the leading-order electroweak fiducial cross section and induce significant shape distortions of differential distributions. The QCD corrections on the other hand are larger ($+24%$) than typical QCD corrections in VBS. This originates from considering the full computation including tri-boson contributions in a rather inclusive phase space. We also provide a leading-order analysis of all contributions to the cross section for ${rm pp to e^{+}e^{-}mu^{+}mu^{-}jj}+X$ in a realistic setup.
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