Tree-level unitarity in Gauge-Higgs Unification

We numerically estimate a scale $\Lambda_{uni}$ at which tree-level unitarity is violated in the $SO(5)\times U(1)_X$ gauge-Higgs unification model by evaluating amplitudes for scattering of the longitudinal W bosons. The scattering amplitudes take larger values in the warped spacetime than in the flat spacetime, and take maximal values when $\theta_H=\pi/2$, where $\theta_H$ is the Wilson line phase along the extra dimension. We take into account not only the elastic scattering but also possible inelastic scatterings in order to estimate $\Lambda_{\rm uni}$. We found that $\Lambda_{\rm uni}\simeq 1.3m_{KK}$ in the warped spacetime, and $\Lambda_{\rm uni}\simeq 140m_{KK}$ in the flat spacetime, where $m_{KK}$ is the Kaluza-Klein mass scale. The tree-level unitarity is violated at $O(1TeV)$ for $\theta_H=\pi/2$ in the former case due to the vanishing WWH coupling.