Abstract
In this paper we present the full NLO QCD + NLO EW corrections to the $Z$-boson pair production in association with a hard jet at the LHC. The subsequent $Z$-boson leptonic decays are included by adopting both the naive NWA and MadSpin methods for comparison. Since the $ZZ+{\rm jet}$ production is an important background for single Higgs boson production and new physics search at hadron colliders, the theoretical predictions with high accuracy for the hadronic production of $ZZ+{\rm jet}$ are necessary. We present the numerical results of the integrated cross section and various kinematic distributions of final particles, and conclude that it is necessary to take into account the spin correlation and finite width effects from the $Z$-boson leptonic decays. We also find that the NLO EW correction is quantitatively nonnegligible in matching the experimental accuracy at the LHC, particularly is significant in high transverse momentum region.
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