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NLO QCD + EW corrections to $ZZZ$ production with subsequent leptonic decays at LHC

In this paper we present the NLO QCD + NLO EW corrections to the $ZZZ$ production with subsequent $Z$-boson leptonic decays at the LHC by adopting the improved narrow width approximation, which takes into account off-shell contributions and spin correlations. Integrated cross sections at $13$, $14$, $33$ and $100~{\rm TeV}$ hadron colliders and various kinematic distributions are presented. Our numerical results show that the jet emission correction accounts for a large part of the total QCD correction, especially in the high energy region. By applying a proper cut to the jet transverse momentum, e.g., $p_{\rm \,T,jet} > p_{\rm \,T,jet}^{\rm \,\,\,cut}=50~{\rm GeV}$, the jet emission correction can be reduced and the cross section is less dependent on the factorization/renormalization scale. This work reveals that both the NLO QCD and the NLO EW corrections are significant. For example, the NLO QCD, NLO EW and NLO QCD + EW relative corrections in the inclusive event selection scheme at the $13~{\rm TeV}$ LHC can reach $63.2\%$, $-9.6\%$ and $47.5\%$, respectively, which means that neither the NLO QCD nor the NLO EW correction is negligible in precision calculations.