NLO QCD + NLO EW corrections to $WZZ$ productions with leptonic decays at the LHC

Precision tests of the Standard Model (SM) require not only accurate experiments, but also precise and reliable theoretical predictions. Triple vector boson production provides a unique opportunity to investigate the quartic gauge couplings and check the validity of the gauge principle in the SM. Since the tree-level predictions alone are inadequate to meet this demand, the next-to-leading order (NLO) calculation becomes compulsory. In this paper, we calculate the NLO QCD + NLO electroweak (EW) corrections to the $W^{\pm}ZZ$ productions with subsequent leptonic decays at the $14~{\rm TeV}$ LHC by adopting an improved narrow width approximation which takes into account the off-shell contributions and spin correlations from the $W^{\pm}$- and $Z$-boson leptonic decays. The NLO QCD+EW corrected integrated cross sections for the $W^{\pm}ZZ$ productions and some kinematic distributions of final products are provided. The results show that both the NLO QCD and NLO EW corrections are significant. In the jet-veto event selection scheme with $p_{T,jet}^{cut} = 50~ {\rm GeV}$, the NLO QCD+EW relative corrections to the integrated cross section are $20.5\%$ and $31.1\%$, while the genuine NLO EW relative corrections are $-5.42\%$ and $-4.58\%$, for the $W^+ZZ$ and $W^-ZZ$ productions, respectively. We also investigate the theoretical dependence of the integrated cross section on the factorization/renormalization scale, and find that the scale uncertainty is underestimated at the LO due to the fact that the strong coupling $\alpha_s$ is not involved in the LO matrix elements.