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Tree-level unitarity in Gauge-Higgs Unification

We numerically estimate a scale $Λ_{uni}$ 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 $θ_H=π/2$, where $θ_H$ 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 $Λ_{\rm uni}$. We found that $Λ_{\rm uni}\simeq 1.3m_{KK}$ in the warped spacetime, and $Λ_{\rm uni}\simeq 140m_{KK}$ in the flat spacetime, where $m_{KK}$ is the Kaluza-Klein mass scale. The tree-level unitarity is violated at $O(1TeV)$ for $θ_H=π/2$ in the former case due to the vanishing WWH coupling.