Tree-level unitarity in Gauge-Higgs Unification Yutaka Sakamura (RIKEN) with Naoyuki Haba (Osaka Univ.) and Toshifumi Yamashita (Nagoya Univ.) December 5, RIKEN seminars arXiv:

1/25 Plan of talk 1.Introduction 2.Set up 3.Weak boson scattering 4.Unitarity violation 5.Summary

2/25 Introduction

3/25 Standard model Higgs boson Electroweak sym. breaking, (perturbative) unitarity ++ e.g.)

4/25 Amplitude 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.

5/25 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,…]

6/25 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 model is non-renormalizable. Tree-level unitarity will be violated at some scale.

7/25 Gauge-Higgs Unification Wilson line phase: HiggsKK modes [Falkowski, Pokorski, Roberts, 2007] main less main Contribution to the saturation of amplitudes

8/25 Set up

9/25 SO(5)xU(1) model on S /Z [Agashe, Contino, Pomarol, 2005] 1 2 tuning q w suppressing T-parameter

10/25 zero-modes Higgs doublet = SO(4) Wilson line phase: Gauge symmetry :

11/25 WWH, ZZH couplings Flat case These are the same as the SM values. Warped case [Hosotani & Y.S., ]

12/25 Weak boson scattering

13/25 Equivalence Theorem longitudinal modewould-be NG boson [Cornwall, Levin & Tiktopoulos, 1974; Lee, Quigg & Thacker, 1977] KK equivalence theorem [Chivukula, Dicus & He, 2002, …]

14/25 Equivalence theorem As an example, we consider.

15/25 Metric Scattering amplitude

16/25 For, each coupling deviates from the SM value. Flat case Warped case [Hosotani & Y.S., 2007]

17/25 The amplitude stops growing when the KK modes start to propagate. In the unit of the KK scale,

18/25 Unitarity violation

19/25 where (S-wave amplitude) Unitarity condition elastic scatteringinvolving KK modes

20/25 Unitarity violation scale L uni

21/25 unitarity cond.

22/25 c.f.

23/25 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.)

24/25 where In the conventional KK expansion,

25/25 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

26/25 Unitarity condition Then we obtain For the 2 →2 channel,

27/25 If we assume that the S-wave component is dominant, we obtain

28/25 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