The Higgs boson mass $m_H=126 $GeV in the $SO(5) times U(1)$ gauge-Higgs unification in the Randall-Sundrum space leads to important consequences. An universal relation is found between the Kaluza-Klein (KK) mass scale $m_{KK}$ and the Aharonov-Bohm
phase $theta_H$ in the fifth dimension; $m_{KK} sim 1350,{rm GeV}/(sin theta_H)^{0.787}$. The cubic and quartic self-couplings of the Higgs boson become smaller than those in the SM, having universal dependence on $theta_H$. The decay rates $H rightarrow gamma gamma, gg$ are evaluated by summing contributions from KK towers. Corrections coming from KK excited states turn out very small. With $theta_H= 0.1 sim 0.35$, the mass of the first KK $Z$ is predicted to be $2.5 sim 6 , $TeV.
When the extra dimensional space is not simply-connected, dynamics of the AB phase in the extra dimension can induce dynamical gauge symmetry breaking by the Hosotani mechanism. This opens up a new way of achieving unification of gauge forces. It lea
ds to the gauge-Higgs unification. The Hosotani mechanism can be established nonperturbatively by lattice simulations, in which measurements of the Polyakov line give a clue.
The gauge bosons and Englert-Brout-Higgs (EBH) boson are unified in the five dimensional RS spacetime. The EBH boson is identified with a part of the fifth dimensional component of the gauge potential. In the SO(5) x U(1) gauge-Higgs unification the
EW symmetry is dynamically broken. The EBH boson, predicted with a mass around 130 GeV, naturally becomes stable so that it appears as missing energy and momentum in collider experiments. Collider signatures such as gauge couplings of quarks and leptons and production of KK gamma and Z are also discussed.
In the $SO(5) times U(1)$ gauge-Higgs unification in the Randall-Sundrum (RS) warped space the Higgs boson naturally becomes stable. The model is consistent with the current collider signatures only for a large warp factor $z_L > 10^{15}$ of the RS s
pace. In order for stable Higgs bosons to explain the dark matter of the Universe the Higgs boson must have a mass $m_h = 70 sim 75$ GeV, which can be obtained in the non-SUSY model with $z_L sim 10^5$. We show that this discrepancy is resolved in supersymmetric gauge-Higgs unification where a stop mass is about $300 sim 320 $GeV and gauginos in the electroweak sector are light.
An SO(5)xU(1) gauge-Higgs unification model in the Randall-Sundrum warped space with top and bottom quarks is constructed. Additional fermions on the Planck brane make exotic particles heavy by effectively changing boundary conditions of bulk fermion
s from those determined by orbifold conditions. Gauge couplings of a top quark multiplet trigger electroweak symmetry breaking by the Hosotani mechanism, simultaneously giving a top quark the observed mass. The bottom quark mass is generated by combination of brane interactions and the Hosotani mechanism, where only one ratio of brane masses is relevant when the scale of brane masses is much larger than the Kaluza-Klein scale (sim 1.5 TeV). The Higgs mass is predicted to be 49.9 (53.5) GeV for the warp factor 10^{15} (10^{17}). The Wilson line phase turns out pi/2 and the Higgs couplings to W and Z vanish so that the LEP2 bound for the Higgs mass is evaded. In the flat spacetime limit the electroweak symmetry is unbroken.
The zero mode of an extra-dimensional component of gauge potentials serves as a 4D Higgs field in the gauge-Higgs unification. We examine QED on $M^4 times S^1$ and determine the mass and potential of a 4D Higgs field (the $A_5$ component) at the two
loop level with gauge invariant reguralization. It is seen that the mass is free from divergences and independent of the renormalization scheme.
