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 $Z \to \bar{f} f \eta$,$Z\to \eta\eta\eta$,$Z\to \eta\gamma$and$Z\to \eta gg$. We examine these decay modes in the parameter space allowed by current experiments, and find that the branching ratios can reach ${10}^{-7}$ for $Z\to \bar{b}b\eta$, ${10}^{-8}$ for $Z\to \bar{\tau }\tau \eta$, and ${10}^{-8}$ for $Z\to \eta \gamma$, which should be accessible at the GigaZ option of the ILC. However, the branching ratios can reach ${10}^{-12}$ for $Z\to \eta \eta \eta$, and ${10}^{-11}$ for $Z\to \eta gg$, which are hardly accessible at the GigaZ option.