Diphoton plus $Z$ production at the ILC at ${\cal O}(\alpha^4)$

Precision measurement for the production of a $Z$-boson in association with two photons is important for investigating the Higgs boson and exploring new physics at the International Linear Collider. It could be used to study the $ZZ\gamma\gamma$ anomalous quartic gauge coupling. In this work we report on our calculation of the full ${\cal O} (\alpha^4)$ contributions to the $e^+e^- \to Z \gamma\gamma$ process in the standard model, and we analyze the electroweak (EW) quantum effects on the total cross section. We investigate the dependence of the $Z\gamma\gamma$ production rate on the event selection scheme and provide distributions for some important kinematic observables. We find that the next-to-leading order (NLO) EW corrections can enhance the total cross section quantitatively from $2.32\%$ to $9.61\%$ when the colliding energy goes up from $250 GeV$ to $1 TeV$, and the NLO EW corrections show obviously a non trivial phase space dependence. We conclude that in studying the signal process $e^+e^- \to ZH \to Z \gamma\gamma$, the background process $e^+e^- \to Z \gamma\gamma$ can be suppressed significantly if we take appropriate kinematic cuts on the final products.