Eq. . Contrary to what has existed in the literature [@Kilian:2004pp; @Kilian:2006eh] for a long time, we found that the coefficient of the antisymmetric $ZH\eta$ vertex $c^{as}_{ZH\eta}$ does not show up until $\ord(\xi^3)$. Based on these results we expect different phenomenological consequences (compared to previous literature) as to the exotic Higgs decay $H\rightarrow Z\eta$ (or $\eta\rightarrow ZH$ if $\eta$ is heavy), the associated production of $h$ and $\eta$ at hadron or lepton colliders, etc. On the other hand, the symmetric $ZH\eta$ vertex already appears at $\ord(\xi)$, and we expect to investigate its effect in a follow-up paper.

The procedure elucidated in this paper can be applied to other models containing a gauged nonlinearly-realized scalar sector as well. From the experience with the SLH we find it important to examine the quadratic part of the Lagrangian in these models, which could contain non-canonically normalized scalar kinetic terms and ‘unexpected’ vector-scalar two-point transitions. Moreover, finding a convenient parametrization for the exponentials in these models could be very helpful in the diagonalization procedure. We expect to investigate these issues and their phenomenological implications in the future.

We thank Kingman Cheung for helpful discussion. This work was supported in part by the Natural Science 