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We re-analyze the $ZH\eta$-vertex with the form$Z_{\mu}(\eta\partial^{\mu}H-H\partial^{\mu}\eta)$, where $H$ is the 125 GeVHiggs boson and $\eta$ is an exotic pseudo-axion, based on the effective fieldtheory (EFT) analysis and choose the simplest little Higgs (SLH) model as anexample. For a pure gauge singlet pseudoscalar $\eta$, after carefully removingall off-diagonal two-point transitions, we show that its coefficient$c_{ZH\eta}$ cannot appear before $\mathcal{O}(\xi^3)$ level, where $\xi$ isthe ratio between the electro-weak scale $v$ and a high scale $f$. The samebehavior arises in the simplest little Higgs (SLH) model, which is quitedifferent from the result that has already existed for a long time.

$ZH\eta$-vertex: EFT Analysis and the Behavior in the SLH Model