Treelevel unitarity in GaugeHiggs Unification Yutaka Sakamura RIKEN

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Tree-level unitarity in Gauge-Higgs Unification Yutaka Sakamura (RIKEN) with Naoyuki Haba (Osaka Univ. )

Tree-level unitarity in Gauge-Higgs Unification Yutaka Sakamura (RIKEN) with Naoyuki Haba (Osaka Univ. ) and Toshifumi Yamashita (Nagoya Univ. ) ar. Xiv: 0908. 1042 December 5, 2009 @ RIKEN seminars

Plan of talk 1. Introduction 2. Set up 3. Weak boson scattering 4. Unitarity

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

Introduction 2

Introduction 2

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

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

Amplitude w/o Higgs unitarity bound w/ Higgs 1 Te. V If the WWH coupling

Amplitude w/o Higgs unitarity bound w/ Higgs 1 Te. V 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

Models with extra dimension EW breaking Higgsless model Boundary conditions along the extra dimension

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

Extra-dimensional model is non-renormalizable. Tree-level unitarity will be violated at some scale. Purpose We

Extra-dimensional model is non-renormalizable. Tree-level unitarity will be violated at some scale. Purpose We numerically estimate • scattering amplitudes for W, Z bosons • a scale at which the tree-level unitarity is violated in the Gauge-Higgs Unification. 6

Gauge-Higgs Unification Wilson line phase: Contribution to the saturation of amplitudes Higgs KK modes

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

Set up 8

Set up 8

SO(5)x. U(1) model on S 1/Z 2 [Agashe, Contino, Pomarol, 2005] tuning qw suppressing

SO(5)x. U(1) model on S 1/Z 2 [Agashe, Contino, Pomarol, 2005] tuning qw suppressing T-parameter 9

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

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

WWH, ZZH couplings Flat case These are the same as the SM values. Warped

WWH, ZZH couplings Flat case These are the same as the SM values. Warped case [Hosotani & Y. S. , 2006 -2007] 11

Weak boson scattering 12

Weak boson scattering 12

Equivalence Theorem [Cornwall, Levin & Tiktopoulos, 1974; Lee, Quigg & Thacker, 1977] longitudinal mode

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

As an example, we consider . Equivalence theorem 14

As an example, we consider . Equivalence theorem 14

Scattering amplitude Metric 15

Scattering amplitude Metric 15

For , each coupling deviates from the SM value. [Hosotani & Y. S. ,

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

In the unit of the KK scale , The amplitude stops growing when the

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

Unitarity violation 18

Unitarity violation 18

Unitarity condition elastic scattering involving KK modes where (S-wave amplitude) 19

Unitarity condition elastic scattering involving KK modes where (S-wave amplitude) 19

Unitarity violation scale Luni 20

Unitarity violation scale Luni 20

unitarity cond. 21

unitarity cond. 21

c. f. 22

c. f. 22

5 D propagator [Gherghetta & Pomarol, 2001] (written by Bessel functions) Advantages We can

5 D propagator [Gherghetta & Pomarol, 2001] (written by Bessel functions) Advantages We can calculate the amplitudes without • the knowledge of the KK mass eigenvalues • summation over infinite KK modes e. g. ) 23

where In the conventional KK expansion, where 24

where In the conventional KK expansion, where 24

Summary • Weak boson scattering in GHU model • Equivalence theorem holds well. •

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 25

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

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

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

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

Comment on Thus, the S-wave amplitude diverges. Taking into account the width of the

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 28