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![Page 1: Introduction to Gauge Higgs unification with a graded Lie algebra 2011. 10. 7 @ Academia Sinica, Taiwan Jubin Park (NTHU) Collaboration with Prof. We-Fu.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649ddb5503460f94ad2bce/html5/thumbnails/1.jpg)
Introduction to Gauge Higgs unification with a graded Lie al-gebra
2011. 10. 7 @ Academia Sinica, Taiwan Jubin Park (NTHU)
Collaboration with Prof. We-Fu Chang
Based on D. B. Fairlie PLB 82,1. G. Bhattacharyya arxiv:0910.5095 [hep-ph]
C. Csaki, J. Hubisz and P. Meade hep-ph/0510275
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Contents• Brief introduction to a difference
between the Higgsless and the Gauge Higgs Unification(GHU) model
Higgsless VS GHU
• Simple examples in the Gauge Higgs unification (GHU) on S1/Z2 - 5D QED
- 5D SU(2) - 5D SU(3)
• Well-known problems in the GHU models• Possible answers for these problems and Goals• Phenomenologically viable GHU models
• A simplest GHU model with a SU(2|1) symmetry. - Lepton coupling
• Summary
2011-10-7
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Alternative models
• - Higgsless no zero modes SM gauge bosons = First excited modes
• - Gauge Higgs Unification SM gauge bosons = Zero modes Needs Higgs mechanism in order to break the EWSB. but there is no Higgs potential in 5D. or Hosotani mechanism. too low Higgs mass (or top quark mass) with VEV which is proportional to 1/R.
2011-10-7Jubin Park @ A. Sinica
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Jubin Park @ A. Sinica
• Simple examples in the Gauge Higgs unification (GHU)
2011-10-7
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2011-10-7Jubin Park @ A. Sinica
5D quantum electrodynamics(QED) on S1/Z2
Model setup
5D GAUGE SYM.
Boundary conditions (BCs)
Periodic BCs
ORBIFOLD BCS
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Jubin Park @ A. Sinica2011-10-7
Kaluza-Klien mode expansion
Remnant gauge symmetry
4D gauge sym. 4D shift sym
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Jubin Park @ A. Sinica2011-10-7
Integrating out fifth dimension
Using a ‘t Hooft gauge.
Propagators
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Jubin Park @ A. Sinica2011-10-7
5D SU(2) example (Non-Abelian case)
Lie algebra valued gauge field
Boundary conditions (BCs)
Projection ma-trix.c
Only diagonal components can have “Zero modes” due to Neumann boundary con-ditions at two fixed pointsGAUGE SYM.
BREAKING
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Jubin Park @ A. Sinica2011-10-7
5D SU(3) example (with 2 scalar dou-blet)
Lie algebra valued gauge field : Gell-Mann mar-tices
Boundary conditions (BCs)
Zero modes.
GAUGE SYM. BREAKING
Branching Rule
*
-
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Jubin Park @ A. Sinica
• Well-known problems in the GHU models
2011-10-7
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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
2011-10-7Jubin Park @ A. Sinica
exp
1tan
3
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Jubin Park @ A. Sinica
• Possible answers for these problems and Goals
2011-10-7
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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)
2011-10-7Jubin Park @ A. Sinica
55
1( )
4 4
aL a F F F F
(2 |1)SU
Abandon the gauge coupling unifi-cation scheme .
Wrong weak mixing angle
R. Coquereaux et.al, CNRSG.~
Burdman and Y.~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].
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• - Using a non-simply connected extra-di-mension ( 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.
2011-10-7Jubin Park @ A. Sinica
256( )L tr F
25~ [ , ]BL A A
Higgs potential
Y. Hosotani, PLB 126, 309, Ann. Phys. 190, 233
N. Manton, Nucl. Phys. B 158, 141
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Jubin Park @ A. Sinica2011-10-7
• One solution for wrong weak mixing an-gle with brane kinetic terms
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Adding to brane kinetic terms
2011-10-7Jubin Park @ A. Sinica
We can easily understand that these terms can give a modification to the gauge couplings without any change of given models.
U(1)
SU(2)
From the effective Lagrangian, we can expect this relation
Similarly, for the U(1) cou-pling
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Final 4D effective La-grangian
2011-10-7Jubin Park @ A. Sinica
g gWeak mixing angle
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 con-stants of given Lie group (or Lie algebra) regardless of volume fac-tor Z if there are no brane kinetic terms in given models.
NO MASS TERM OF THE HIGGS
BECAUSE OF HIGHER DI-MENSIONAL GAUGE SYM-
METRY
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HIGGS POTENTIAL,
| | ,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,
NAMBU-GOLDSTONE BOSON MODES ~
MASSLESS (FLAT DIRECTION)
RADIAL MODES ~MASSIVE
2011-10-7Jubin Park @ A. Sinica
4H DM g v 4
1
DW
g vM
Z
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Goals
• Stability of the electroweak scale (from the quadratic divergences – Gauge hierarchy problem)
• Higgs potential
- to trigger the electroweak symmetry breaking
• Correct weak mixing
2011-10-7Jubin Park @ A. Sinica
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Jubin Park @ A. Sinica
• Phenomenologically viable GHU modelsPhenomenologically viable GHU models
2011-10-7
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Jubin Park @ A. Sinica
• A simplest GHU model with a SU(2|1) symmetry.
2011-10-7
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Jubin Park @ A. Sinica2011-10-7
Model setup : A pure Yang-Mills theory on 6D
Covariant derivative and Field strength
SU(2)
U(1)
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Jubin Park @ A. Sinica2011-10-7
Covariant derivative of the scalar
Hyper charge
Effective kinetic term in 4D
POTENTIAL OF SCALAR
KINETIC TERM
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Jubin Park @ A. Sinica
However, the Higgs mechanism can not happen due to the sign of quadratic term. That is to say, the photon remains massless.
2011-10-7
K = 2 CASE
1. Hyper charge of scalar = -3
° Embedding SU(3) GHU without diagonal compo-nents of zero modes of A5 and A6
(2) (1) (3)SU U SU
(2) (1) (3)SU U SU
3. Mixing between diagonal generators
2. A electroweak mixing angle
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Jubin Park @ A. Sinica2011-10-7
• K = -2 CASE This is not a Lie algebra ( Traceless cond.)
1. Hyper charge of scalar = +1
2. We can have the same relations in the model, like SM has.
?
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Jubin Park @ A. Sinica2011-10-7
? No zero trace condition because of K=-2, -1-1 + k ≠0
Supertracetr(a) tr(b)
Supertraceless
=0
4 5 6 7, , , can satisfy usual SU(2) and U(1) Lie algebra commutators
4 5 6 7, , ,
can satisfy anticommutators(ACs),and these ACs generatesusual Lie transformation. (Closed)
Z2 graded Lie algebra - SU(2|1) V. G. Kac, Commum. Math. Phys. 53, 31
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Jubin Park @ A. Sinica2011-10-7
An general gauge field that couples to the element T of SU(2|1)
Infinitesimal transformation under T element of SU(2|1)
where
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Jubin Park @ A. Sinica2011-10-7
The field strength F in this model with the SU(2|1)
The Kinetic term is
The F46, F55, and F66 terms are
Note that A is not neither hermitian nor antisymmetric !!!!!!!!
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Jubin Park @ A. Sinica2011-10-7
Finally we can have this interesting(?) potential,
Unlike previous Lie gauge,
this model can give correct sign of quadratic term to the Higgs po-tential in order to trigger Higgs mechanism, and also give correct hypercharge +1 to the scalar particle.
After the Higgs mechanism,
From the VEV, a mass of the Higgs is
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Summary
• The graded Lie algebra in the GHU scheme can give the correct SM-like Lagrangian at low energy .
- Correct weak mixing angle.
- Needed Higgs potential for Higgs mechanism. - Not too small mass of the Higgs.
2011-10-7Jubin Park @ A. Sinica