Introduction to Gauge Higgs unification with a graded Lie algebra

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

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

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

Jubin Park @ A. Sinica

• Simple examples in the Gauge Higgs unification (GHU)

2011-10-7


5D quantum electrodynamics(QED) on S1/Z2

Model setup5D GAUGE SYM.

Boundary conditions (BCs)

Periodic BCsORBIFOLD

BCS

Jubin Park @ A. Sinica2011-10-7

Kaluza-Klien mode expansion

Remnant gauge symmetry

4D gauge sym. 4D shift sym


Integrating out fifth dimension

Using a ‘t Hooft gauge.Propagators


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


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

*

-


• Well-known problems in the GHU models

2011-10-7

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


exp1tan3


• Possible answers for these problems and Goals

2011-10-7

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)


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].

• - 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.


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


• One solution for wrong weak mixing angle with brane kinetic terms

Adding to brane kinetic terms


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

Final 4D effective La-grangian


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

HIGGS POTENTIAL,

| | ,After the Higgs obtains H v 42 2 ,2D

H Wg 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


4H DM g v 4

1

DW

g vMZ

Goals• Stability of the electroweak scale

(from the quadratic divergences – Gauge hierarchy problem)

• Higgs potential

- to trigger the electroweak symmetry breaking

• Correct weak mixing



• Phenomenologically viable GHU modelsPhenomenologically viable GHU models

2011-10-7


• A simplest GHU model with a SU(2|1) symmetry.

2011-10-7


Model setup : A pure Yang-Mills theory on 6D

Covariant derivative and Field strength

SU(2)

U(1)


Covariant derivative of the scalar

Hyper charge

Effective kinetic term in 4D

POTENTIAL OF SCALAR

KINETIC TERM


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


• 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.

?


? 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


An general gauge field that couples to the element T of SU(2|1)

Infinitesimal transformation under T element of SU(2|1)

where


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 !!!!!!!!


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

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.


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Introduction to Gauge Higgs unification with a graded Lie algebra

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