Two Higgs doublets model in gauge-Higgs unification
framework
2013. 6. 8 @ Yonsei University
Jubin Park (SNUT)
Collaboration with Prof. We-Fu Chang, Prof. Sin Kyu Kang
A fundamental scalar field (Higgs) is introduced to explain spontaneous symmetry breaking of gauge group of electroweak symmetry.
The same field is also responsible for masses of all matter fields through Yukawa interactions.
Standard Model
Jubin Park @ 2013 NRF WORKSHOP
The Higgs potential is written by HAND.
2 2 2(| | )HiggsV v
So the Higgs sector is very sensitive to the UV scale of the theory
Without symmetry protection,
2 2Higgsm
Moreover,
Unknown origin
Hierarchy Problem
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Unification of gravity (s=2) & electromagnetic (s=1)
Kaluza-Klein gravity theory
Unified theory of gauge (s=1) & Higgs (s=0)
Gauge-Higgs unification
4D space-time 4D gauge-field Higgs 5D gauge field extra dimension
The pioneer works of GHU :
・ N.S. Manton, Nucl. Phys. 58(’79)141.
・ Y. Hosotani, Phys. Lett. B126 (‘83) 309 ``Hosotani mechanism”
Jubin Park @ 2013 NRF WORKSHOP
Higher dimensional Gauge Theory
Jubin Park @ 2013 NRF WORKSHOP
00 01 02 03 0
10 11 12 13 1
20 21 22 23 2
30 3231 31 3
0 31 2
MN
g g g g A
g g g g A
G g g g g A
g gg g A
A AA A
Kaluza-Klein gravity theory
Gauge-Higgs unification
0
1
2
3
M
A
A
A A
A
WMbmL QCD Weak scale
Hadronization scale
B physics scale
tm
200 MeV 5 GeV 80 GeV 172 GeV 10 TeV
Energy scales
P lM
1 TeV
Compactification scale
1 / CR
?
LTheory cutoff scale
GUTMMajoranaM
10 ^19 GeV10^15 ~ 10 ^17 GeVPlanck scale-
strong gravity
GUT- coupling
unification
Heavy right-handed
Majorana for Seesaw
Mechanism
Jubin Park @ 2013 NRF WORKSHOP
Jubin Park @ 2013 NRF WORKSHOP
5D quantum electrodynamics(QED) on S1/Z2
Model setup
Boundary conditions (BCs)
Jubin Park @ 2013 NRF WORKSHOP
5D SU(2) example (Non-Abelian case)
Lie algebra valued gauge field
Boundary conditions (BCs)
Only diagonal components can have “Zero modes” due to Neumann boundary conditions at two fixed points
Jubin Park @ 2013 NRF WORKSHOP
5D SU(3) example (with 2 scalar doublets)
Lie algebra valued gauge field
Boundary conditions (BCs)
We only focus on the zero modes,
After we integrate out fifth dimension,
0 0 0 00
5( )a abc b cDgZ Z ZF A A fZ
A Z A
And rescale the gauge field,
Jubin Park @ 2013 NRF WORKSHOP
Adding to brane kinetic terms
Jubin Park @ 2013 NRF WORKSHOP
We can easily understand that these terms can give a modification to the gauge couplings without any change of given models.
From the effective Lagrangian, we can expect this relation
Similarly, for the U(1) coupling
Final 4D effective Lagrangian
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g g
1 2
* Note that the value of tangent angle
for weak mixing angle is 3 0.whenc c
This number is completely fixed by the analysis of structure constants of given Lie group (or Lie algebra) regardless of volume factor Z if there are no brane kinetic terms in given models.
Well-known problems
• Wrong weak mixing angle( , , )
• No Higgs potential (to trigger the EWSB). - may generate too low Higgs mass (or top quark) even if we use quantum corrections to make its potential.
• Realistic construction of Yukawa couplings
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exp
1tan
3
Possible answers for these problems
- Brane kinetic terms
- Violation of Lorentz symmetry ( SO(1,4) -> SO(1,3) )
- Graded Lie algebra (ex. )
- Using a non-simple group. an anomalous additional U(1) (or U(1)s)
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55
1( )
4 4
aL a F F F F
(2 |1)SU
R. Coquereaux et.al, CNRSG.~
Burdman and .~Nomura, Nucl. Phys. B656, 3. (2003) : arXiv:hep-ph/0210257].
I. Antoniadis, K. Benakli and M. Quiros, New J. Phys. 3, 20 (2001) [arXiv:hep-th/0108005].
• - Using a non-simply connected extra-dimension ( the fluctuation of the AB type phase – loop quantum correction)
- Using a 6D (or more) pure gauge theory.
- Using a background field like a monopole in extra dimensional space.
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256( )L tr F
25~ [ , ]BL A A
Y. Hosotani, PLB 126, 309, Ann. Phys. 190, 233
N. Manton, Nucl. Phys. B 158, 141
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1-loop generated Higgs potential
Cosmological Constant for more generalized GHU model
means KK states are ~ 200 GeV
OR
Monopole-like background
-Forbidden by higher dimensional gauge symmetry
22 2 4
0 5 52eff tree
gV V V A g A
R
22 2
5 2 2
1/ 1/
gA R
R g
From commutator terms in 6d or higher dimensions (>6d).
| | ,After the Higgs obtains H v 42 2 ,2D
H W
g vM v M
24,2Dg
Finally, we can get this relation ( with brane Kinetic terms ),
We can rewrite the equation with previous relation,
Jubin Park @ 2013 NRF WORKSHOP
4H DM g v 4
1
DW
g vM
Z
So our goals are
• Stability of the electroweak scale (from the quadratic divergences – Gauge hierarchy problem)
• Higgs potential
- to trigger the electroweak symmetry breaking
• Correct weak mixing
Jubin Park @ 2013 NRF WORKSHOP
Jubin Park @ 2013 NRF WORKSHOP
• We focus on the GHU models in 6D with Brane kinetic terms
Therefore,
Jubin Park @ 2013 NRF WORKSHOP
The Lie algebra valued function in complex coordinate (5D + 6D)
Field assignments in terms of the Higgs doublets
Higgs doublet
The GHU models in 6D
Jubin Park @ 2013 NRF WORKSHOP
General results of the commutators
Relations for the indices of structure constants
One constraint
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Three conditions for the existence of the doubletsFirst condition for the triangle :
Second condition for the triangle :
VVVVVVVVVVVVVV
So these three vectors should make
an equilateral triangle in the root space
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Allowed angle between beta and gamma vectors and corresponding possible groups and triangles through Lie algebra
Third condition for the triangle :
Not allowed ratios between lengths of two root vectors by the Lie algebra
Jubin Park @ 2013 NRF WORKSHOP
Typical form of the GHU potential
From commutator terms in 6d or higher dimensions (>6d).
1-loop generated Higgs potential
with the identification 3g^2=g’^2
Jubin Park @ 2013 NRF WORKSHOP
1-loop generated Higgs potential
Cosmological Constant for more generalized GHU model
means KK states are ~ 200 GeV
OR
Monopole-like background
-Forbidden by higher dimensional gauge symmetry
22 2 4
0 5 52eff tree
gV V V A g A
R
22 2
5 2 2
1/ 1/
gA R
R g
From commutator terms in 6d or higher dimensions (>6d).
Again, structure of the Higgs potential
in the GHUm
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Characteristic of all Lie groups
in the GHUm with 2HDs
exp
1tan
3 2
expsin 0.22292
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Modified mass relation between the W boson and the Higgs masses
This factor reduces the proportional constant number.
So almost GHU models (except exceptional groups) in the decoupling limit can not easily escape the Well-known LEP bound, 114.4 GeV .
Jubin Park @ 2013 NRF WORKSHOP
Heavy u(1) prime condition :
Needed c2* number to go to the heavy U(1) prime scenario
Numerical results for the light Higgs particle in the decoupling limit.
All masses are smaller than 114.4 GeV.
Jubin Park @ 2013 NRF WORKSHOP