Tree-level unitarity in Gauge-Higgs Unification Yutaka Sakamura (KEK) with Naoyuki Haba (Osaka Univ.) and Toshifumi Yamashita (Nagoya Univ.) January 20, Osaka Univ. arXiv:

1/21 Introduction

2/21 Standard model Higgs boson Electroweak sym. breaking, (perturbative) unitarity ++ e.g.)

3/21 1 TeV Unitarity bound w/o Higgs w/ Higgs If the WWH coupling vanishes, the Higgs boson cannot contribute to the unitarization. This occurs in the Gauge-Higgs Unification models in the warped spacetime.

4/21 We numerically estimate scattering amplitudes for W, Z bosons a scale at which the tree-level unitarity is violated in the Gauge-Higgs Unification. Purpose Extra-dimensional models are non-renormalizable. Tree-level unitarity will be violated at some scale. From naïve dimensional analysis,

5/21 EW breaking Boundary conditions along the extra dimension Higgsless model Unitarity is recovered by KK gauge bosons Gauge-Higgs Unification Unitarity is recovered by KK gauge bosons and zero-mode of [Csaki, et.al, 2003] Higgs Models with extra dimension [Fairlie; Manton, 1979; Hosotani, 1983,…]

6/21 Gauge-Higgs Unification in warped spacetime Wilson line phase: HiggsKK modes [Falkowski, Pokorski, Roberts, 2007] main less main Contribution to the saturation of amplitudes

7/21 Set up

8/21 SO(5)xU(1) model on S /Z [Agashe, Contino, Pomarol, 2005] 1 2 tuning q w suppressing T-parameter Planck brane TeV brane

9/21 zero-modes Higgs doublet = SO(4) Wilson line phase: Gauge symmetry : determined by quantum effect

10/21 WWH, ZZH couplings Flat case These are the same as the SM values. Warped case [Hosotani & Y.S., ]

11/21 Weak boson scattering

12/21 Equivalence theorem As an example, we consider. Equivalence Theorem

13/21 Metric Scattering amplitude

14/21 For, each coupling deviates from the SM value. Flat case Warped case [Hosotani & Y.S., 2007]

15/21 The amplitude stops growing when the KK modes start to propagate. In the unit of the KK scale,

16/21 Unitarity violation

17/21 where (S-wave amplitude) Unitarity condition elastic scatteringinvolving KK modes

18/21 Unitarity violation scale L uni

19/21 unitarity cond.

20/21 c.f. unitarity cond.

21/21 Summary Weak boson scattering in GHU model Equivalence theorem holds well. Amplitudes have large -dependence in the warped spacetime. Tree-level unitarity is violated at

22/21 5D propagator Advantages the knowledge of the KK mass eigenvalues summation over infinite KK modes We can calculate the amplitudes without [Gherghetta & Pomarol, 2001] (written by Bessel functions) e.g.)

23/21 where In the conventional KK expansion,

24/21 Unitarity condition Then we obtain For the 2 →2 channel,

25/21 If we assume that the S-wave component is dominant, we obtain

26/21 Comment on Thus, the S-wave amplitude diverges. Taking into account the width of the W boson, the divergence at is smeared out. translated into a cut-off for f