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

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 GeV Higgs boson and $\eta$ is an exotic pseudo-axion, based on the effective field theory (EFT) analysis and choose the simplest little Higgs (SLH) model as an example. For a pure gauge singlet pseudoscalar $\eta$, after carefully removing all off-diagonal two-point transitions, we show that its coefficient $c_{ZH\eta}$ cannot appear before $\mathcal{O}(\xi^3)$ level, where $\xi$ is the ratio between the electro-weak scale $v$ and a high scale $f$. The same behavior arises in the simplest little Higgs (SLH) model, which is quite different from the result that has already existed for a long time.