NLO QCD and electroweak corrections to $WWW$ production at the LHC

In this paper we calculate the next-to-leading order (NLO) QCD $[{\cal O}(\alpha_s \alpha^3)]$ and electroweak (EW) $[{\cal O}(\alpha^4)]$ corrections to the $WWW$ production at the LHC and deal with the subsequent leptonic decays from $W$ bosons by adopting an improved narrow-width approximation which takes into account the spin correlation and finite-width effects. The NLO QCD correction from the real jet radiation is discussed, which significantly enhances the production rate, particularly in the high-energy region. We also provide the integrated cross section for the $WWW$ production and various kinematic distributions of final products at the QCD+EW NLO. We find that the convergence of the perturbative QCD description can be improved by applying a hard jet veto in the event selection, but this jet veto would introduce a new source of theoretical uncertainty. The pure NLO QCD relative correction to the integrated cross section for the $W^+W^-W^+$ production at the $14~{\rm TeV}$ LHC is on the order of $30\%$ in the jet-veto event selection scheme with $p^{\, {\rm cut}}_{T,\, {\rm jet}} = 50~{\rm GeV}$, while the genuine NLO EW relative correction can reach about $15\%$ in the inclusive event selection scheme. Our numerical results show that both the NLO QCD and NLO EW corrections should be taken into consideration in precision predictions.