  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 ${\rm pp\to
e^{+}e^{-}\mu^{+}\mu^{-}jj}+X$ at orders $\mathcal{O}(\alpha^7)$ and
$\mathcal{O}(\alpha_s\alpha^6)$ 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.

High Energy Physics - Phenomenology