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

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