Tree-level unitarity in Tree-level unitarity in Gauge-Higgs UnificationGauge-Higgs Unification

Yutaka Sakamura (KEK) with Naoyuki Haba (Osaka Univ.)

and Toshifumi Yamashita (Nagoya Univ.)

January 20, 2010 @ Osaka Univ.

arXiv:0908.1042

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IntroductionIntroduction

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Standard modelHiggs boson Electroweak sym. breaking,

(perturbative) unitarity

+ +

e.g.)

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1 TeV

Unitaritybound

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 modelsin the warped spacetime.

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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,

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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,…]

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Gauge-Higgs Unificationin warped spacetime

Wilson line phase:

Higgs KK modes

[Falkowski, Pokorski, Roberts, 2007]

main

less

less

main

Contribution to the saturation of amplitudes

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Set upSet up

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SO(5)xU(1) model on S /Z[Agashe, Contino, Pomarol, 2005]

1 2

tuning w

suppressing T-parameter

Planck brane TeV brane

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zero-modes

Higgs doublet

=

SO(4)

Wilson line phase:

Gauge symmetry :

determined by quantum effect

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WWH, ZZH couplingsFlat case

These are the same as the SM values.

Warped case [Hosotani & Y.S., 2006-2007]

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Weak boson scatteringWeak boson scattering

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Equivalence theorem

As an example, we consider .

Equivalence Theorem

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Metric

Scattering amplitude

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For , each coupling deviates from the SM value.

Flat case

Warped case

[Hosotani & Y.S., 2007]

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The amplitude stops growing when the KK modes start to propagate.

In the unit of the KK scale ,

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Unitarity violationUnitarity violation

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where

(S-wave amplitude)

Unitarity condition

elastic scattering involving KK modes

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Unitarity violation scale uni

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unitarity cond.

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c.f.

unitarity cond.

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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

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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.)

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where

where

In the conventional KK expansion,

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Unitarity condition

Then we obtain

For the 2 →2 channel,

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

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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