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RS model

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RS model

Jubin Park, 朴朱彬

(PRL 83,3370 and PRL 83,4690 Lisa Randall, Raman Sundrum)

2005.5.13 Yonsei Univ.

• Introduction

•A brief review of General Relativity

•Model setup

•Classical Solution

•Physical implication

•Summary

•If we live in fundamentally higher dimensions, then effective four-dimensional Planck scale is determined by the fundamental higher dimensional energy scale and geometryof the extra dimensions.

•So we will present a higher dimensional model (is called a RS model) to solve the hierarchy problem.

* the hierarchy problem is why the typical EW (electroweak) energy scale associated with SSB (spontaneous symmetry breaking) is so much much smaller than the Planck energy scale.

Geometry

Connection

Cuverture

Energy

Mass and Energy

=

* Relativistic units (c=1, G=1)

• Ricci tensor

• Ricci scalar

• Einstein tensor

• Energy-momentum tensor

• Einstein Equation

• First We launch out our space-time with orbifold structure.

Periodicity

Identify

•We assume that the metric is not factorizable but the four-dimensional metric is multiplied by a warp factor. This warp factor only depends on fifth extra dimension.

A figure of space in RS model

Visible brane

Planck brane

Is the five-dimensional metric

•The classical action is

Where is a five dimensional cosmological constant and

and are constant “vacuum energy” which acts as a source of gravity.

•We can obtain five-dimensional Einstein’s equations from a previous classical action by a variational principle.

5d Einstein’s equation

•We assume that there exist a solution of Einstein’s equations satisfying the 4d Poincare invariance.

where

Connections

Riemann tensors

Ricci tensor

Ricci scalar

With this ansatz, the Einstein’s equations reduce to

(1) M=N=5

(2) M=N=0,1,2,3

Therefore we need to find a solution of .

The solution of

From this, if is a solution of we can see the relation between , and

Finally we can get a exact metric form of our model.

•We can extract physics in which we interested with a four-dimensional effective field theory.

5D theory

4D effective theory

Physical graviton of 4d effective theory

• Let me focus on curvature term

Where denotes the four-dimensional Ricci scalar and is a 4d metric.

After integration, we can get

depends on only weakly in in the large limit.

•As you compare the metric of 4d effective theory with our 5d metric you can see relations between them

•For example, Consider a fundamental Higgs field

Substituting our metric relation into this action yields

After wave-function renormalization, , we obtain

• Any mass parameter on visible 3-brane in the fundamental higher dimensional theory will correspond to a physical mass.

If and ,

then

• In our model we introduce two 3-branes (Plank brane and Visible brane) in extra fifth dimension and the visible brane contains the standard model fields.

• There is a warped geometry or warped factor which generates the hierarchy.

• The weak scale is generated from the Planck scale through an exponential hierarchy.

• In this model flavor violation and proton decay remain important challenges.

the hierarchy problem is the big question why the typical energy scale associated with the electroweak symmetry breaking - roughly, the typical size of all masses of elementary particles - is so much ( times) smaller than the Planck energy. More technically, the question is why the Higgs boson is so much lighter than the Planck mass, although one would expect that the large (quadratically divergent) quantum contributions to the Higgs boson mass would inevitably make the mass huge, comparable to the Planck mass.

–Brainy Encyclopedia.

Kaluza – Klein Graviton

After an integration by part

KK expansion