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We report on the precision predictions for the $e^+e^-\to ZZ\gamma$ processincluding $Z$-boson leptonic decays at the ILC in the standard model (SM). Thecalculation includes the full next-to-leading (NLO) electroweak (EW)corrections and high order initial state radiation (h.o.ISR) contributions. Wefind that the NLO EW corrections heavily suppress the LO cross section, and theh.o.ISR effects are notable near the threshold while become small in highenergy region. We present the LO and the NLO EW+h.o.ISR corrected distributionsof the transverse momenta of final $Z$-boson and photon as well as the $Z$-pairinvariant mass, and investigate the corresponding NLO EW and h.o.ISR relativecorrections. We also study the leptonic decays of the final $Z$-boson pair byadopting the {\sc MadSpin} method where the spin correlation effect isinvolved. We conclude that both the h.o.ISR effects and the NLO EW correctionsare important in exploring the $ZZ\gamma$ production at the ILC.

Precision study on $ZZ\gamma$ production including $Z$-boson leptonic decays at the ILC