Electroweak radiative corrections to

We provide and discuss the precision predictions for the $$W^+W^-\gamma $$ production at the ILC including the full electroweak (EW) one-loop corrections and high order initial state radiation (ISR) contributions in the standard model. The dependence of the leading order (LO) and EW corrected cross sections on the colliding energy is investigated. We find that the EW correction suppresses the LO cross section significantly, and the ISR effect beyond $${\mathcal O}(\alpha )$$ is important near the threshold, but it is negligible in the high energy region. We provide the LO and EW corrected distributions of the transverse momenta and rapidities of final $$W^-$$-boson and photon as well as the $$W$$-pair invariant mass. From the various kinematic distributions, we find that the EW correction strongly depends on the final state phase space. We investigate the leptonic decays of the final W-boson pair by adopting the narrow width approximation, and we find that the final produced photon and leptons can be well separated from each other