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
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