$W$ boson mass in gauge-Higgs unification

The $W$ boson mass $m_W$ in the GUT inspired $SO(5)\times U(1)\times SU(3)$ gauge-Higgs unification in the Randall-Sundrum (RS) warped space is evaluated. The muon decay ${\mu }^{-}\to e^{-}{\bar{\nu }}_e{\nu }_{\mu }$ proceeds by the exchange of not only the zero mode of the $W$ boson $(W^{(0)}$) but also Kaluza-Klein (KK) excited modes $W^{(n)}$ and $W_R^{(n)}$ ($n\ge 1$) at the tree level. The anti-de Sitter curvature of the RS space also affects the relationship among the gauge couplings and the ratio of $m_W$ to the $Z$ boson mass $m_Z$. The $W$ couplings of leptons and quarks also change. With the given KK mass scale $m_{\mathrm{K}\mathrm{K}}$ the range of the Aharonov-Bohm phase ${\theta }_H$ in the fifth dimension is constrained. For $m_{\mathrm{K}\mathrm{K}}=13$TeV, $0.085\lesssim {\theta }_H\lesssim 0.11$ and $80.381\mathrm{G}\mathrm{e}\mathrm{V}\lesssim m_W\lesssim 80.407\mathrm{G}\mathrm{e}\mathrm{V}$. The predicted value of $m_W$ for $13\mathrm{T}\mathrm{e}\mathrm{V}\le m_{\mathrm{K}\mathrm{K}}\le 20\mathrm{T}\mathrm{e}\mathrm{V}$ lies between $m_W^{\mathrm{S}\mathrm{M}}=80.354\pm 0.007$GeV in the standard model and $m_W^{\mathrm{C}\mathrm{D}\mathrm{F}}=80.4335\pm 0.0094$GeV, the value reported by the CDF collaboration in 2022.