Rare Z-decay into light pseudoscalar bosons in the simplest little Higgs model

The simplest little Higgs model predicts a light pseudoscalar boson $eta$ and opens up some new decay modes for $Z$-boson, such as $Z to bar{f} f eta$, $Zto etaetaeta$, $Zto etagamma$ and $Zto eta gg$. We examine these decay modes in the parameter space allowed by current experiments, and find that the branching ratios can reach $10^{-7}$ for $Zto bar{b}beta$, $10^{-8}$ for $Zto bar{tau}taueta$, and $10^{-8}$ for $Zto etagamma$, which should be accessible at the GigaZ option of the ILC. However, the branching ratios can reach $10^{-12}$ for $Zto etaetaeta$, and $10^{-11}$ for $Zto eta gg$, which are hardly accessible at the GigaZ option.