Tree-level unitarity in
Gauge-Higgs Unification
Yutaka Sakamura (KEK)
with Naoyuki Haba (Osaka Univ.)
and Toshifumi Yamashita (Nagoya Univ.)
arXiv:0908.1042
January 20, 2010 @ Osaka Univ.
Introduction
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Standard model
Higgs boson
Electroweak sym. breaking,
(perturbative) unitarity
e.g.)
+
+
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w/o Higgs
Unitarity
bound
w/ Higgs
1 TeV
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.
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Extra-dimensional models are non-renormalizable.
Tree-level unitarity will be violated at some scale.
From naïve dimensional analysis,
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.
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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
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Gauge-Higgs Unification
in warped spacetime
Wilson line phase:
Contribution to the saturation of amplitudes
Higgs
KK modes
main
less
less
main
[Falkowski, Pokorski, Roberts, 2007]
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Set up
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SO(5)xU(1) model on S1/Z 2
[Agashe, Contino, Pomarol, 2005]
Planck brane
TeV brane
tuning qw
suppressing T-parameter
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Gauge symmetry :
zero-modes
=
SO(4)
Higgs doublet
Wilson line phase:
determined by quantum effect
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WWH, ZZH couplings
Flat case
These are the same as the SM values.
Warped case
[Hosotani & Y.S., 2006-2007]
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Weak boson scattering
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Equivalence Theorem
As an example, we consider
.
Equivalence theorem
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Scattering amplitude
Metric
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For
, each coupling deviates from the SM value.
[Hosotani & Y.S., 2007]
Flat case
Warped case
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In the unit of the KK scale
,
The amplitude stops growing when the KK modes
start to propagate.
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Unitarity violation
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Unitarity condition
elastic scattering
involving KK modes
where
(S-wave amplitude)
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Unitarity violation scale Luni
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unitarity cond.
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unitarity cond.
c.f.
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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
[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.)
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where
In the conventional KK expansion,
where
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Unitarity condition
For the 2 →2 channel,
Then we obtain
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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 f
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