Extra Dimensions (XD)

Xiaozhou zhou9/25/2007

Outline

What does extra dimension mean?Why extra dimensions?Why can’t we see extra dimensions?The history of extra dimension theoriesHow to discribe extra dimensionsModern extra dimension models(ADD

and RS)

What extra XD? Extra dimension: Extra degrees of freedom which can be interpreted as space-

time coordinates We only consider space-like XD here, i.e., there are always

only one time dimension. Suppose we have one more time-like dimension, invariant

mass:

We can find some mechanism to get rid of tacyon, but that’s

more complicated.

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p25D = p0 p0 − p.p + p5 p5 ⇒

m24D = p2

4 D = p25D − p5 p5!≥ 0

m24D < 0⇒ ' tacyon'

⇒ causality violation

Why XD?

The simplest answer: Theorists can always guess anything that’s not ruled out

by experiments.

Why isn’t it a waste of life? It’s required by string theory It can give rise to some interesting mechanisms

which may solve the theoretical problems in Standard Model

It may unify different symmetries in 4D with more fundamental higher dimensional symmetry

It can be a type of emergent phenomenon with some kind of duality (typically AdS/CFT)

Why can’t we see XD? If there are extra dimensions, why can’t we see them?

Or more formally, why is our physics all 4 dimensional-like?

There are two ways to “hide” extra dimensions: Compactification The 4D(3+1) world is infinite,while extra dimensions are compacted with a very small scale which can not be detectedby low energy experiments. Brane-world scenario Our 3D world is localized on some subspace of the

whole universe called D-brane. High energy is needed to escape to extra dimensions.

The ‘Ancient’ XD theoryKaluza(1919)-Klein(1926) Model Added one extra spacial dimension (compacted to S1) to Einstein’s

General Relativity equation 4D gravity and electromagnitic field are unified in a 5D gravity theory

Gauge symmetry of EM field is interpreted as the traslational symmetry of the 5th dimension. The periodical boundary condition of the 5th dimension gives rise to quantization of charge

A severe problem: the coupling strength of the emergent 4D gravity and EM field are

exactly the same, which is conflicting with even common sense XD idea was discarded for decades until……

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5D gravity : GMN

decomposition− >

4D gravity(tensor) : gμν = Gμν

4D gauge(vector) : Aμ = G5μ

4D radion(scalar) : R = G55

⎧

⎨ ⎪

⎩ ⎪

String Theory The fundamental blocks in string theory are one

dimensional strings The exact number of space-time dimensions can be

calculated by considering the consistence of the theory (Eg, photons should be massless) Bosonic string(only contains bosons): 26 dimensions Superstring(with supersymmetry): 10 dimensions M-theory(with supergravity): 11 dimensions

• By using topological tools to constrain spacetime structure and compactification, we can get rid of the old problems

• The big embarrassment: No way to go down!

Modern XD theories With some borrowed ideas from string

theory but much closer to the real world. Effective theory with energy scale higher

than Standard Model but much lower than Plank scale (10^19GeV). May be considered as low energy approximation of string theory, or just some independent models with unknown UV completion.

Rich phemomena: Much easier to give predictions to experimental signals within the range of Large Hadron Collider(LHC), which is about several TeV. May be tested soon.

Variace of models: ADD, RS, UED……

How to describe XD? Geometric description of spacetime:

Kaluza-Klein(KK) Decomposition

Masses of KK tower:

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S(4 +n )D = d4 x∫ dn y | G |∫ L(Φ,DM Φ;Ψ,DM Ψ;AM ,DM AN ;GMN )

Eg : 5D EM field

S5D = d4 x dy GGMNGLK FMLFNK∫∫FML = DM AL − DL AM ,DM AL = ∂M AL − ΓML

N AN

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5D EM Field : AM (x μ , y) = AM(n )(x μ ) f (n )(y)

n=−∞

∞

∑

Flat XD : f (n )(y) = e iny / R

Gauge fixing :

A5(x μ , y) = A5(0)(x) → Scalar

Aμ (x μ ,y) = Aμ(0)(x) + (

n=1

∞

∑ Aμ(n )(x)e inφ + h.c.) → KK tower

⎧

⎨ ⎪

⎩ ⎪

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mn ~n

R

4D Effective Field Theory Higher dimension <=> Higher energy scale To go back to 4D effective theory, we need these steps:

1. Throwing away high order modes, only keep the modes with masses in our experimental range

2. Integrating out the extra dimensions3. Renormalizing the wave function4. Redefining the coupling constant

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Eg :Scalar field(flat 5D)

S ~E<<R −1

2πR d4 x∫ ∂μϕ(0)(x)∂ μϕ (0)(x) − m2ϕ (0)(x)ϕ (0)(x) +

λ

4ϕ (0)4 (x)

⎡ ⎣ ⎢

⎤ ⎦ ⎥

redefine :ϕ (0) → ′ ϕ = 2πRϕ (0),λ → ′ λ = λ 2πR

S4 Deff ~ d4 x ∂μ ′ ϕ (x)∂ μ ′ ϕ (x) − m2 ′ ϕ (x) ′ ϕ (x) +′ λ

4′ ϕ 4 (x)

⎡ ⎣ ⎢

⎤ ⎦ ⎥∫

Large XD ADD model: Nima Arkani-Hamed, Savas Dimopoulos & Gia Dvali(1998) Set-up:

Number of extra dimensions: n Flat extra dimensions compacted as a circle with radius

R Fermions and gauge fields are confined to our 4D world Gravity can propagate freely in all the dimensions

• Solving hierachy problem: Hierachy problem: Why is gravitational interaction much

weaker than strong and electroweak interactions? Solution in ADD: The coupling of gravity is about the

same order of oher interactions, but it’s diluted by extra dimensions.

Large XD

• Experimental predictions Deviation from Newton’s gravity at scale smaller than R How small? n=1 -> R~10^18km (scale of solar system) - ruled out! n=2 -> R~1mm - on the edge of experiment KK graviton spectrum on the collider small mass splitting(almost continuous spectrum): Many contraints come from astronomy and cosmology

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n = 2 → Δm =1

R~ 10−4 eV

Warped XD RS model: Lisa Randall & Raman Sundrum (1999) RS1 Set-up:

One extra dimension with non-trival warp factor Fermions and gauge fields are on the IR 3-brane Gravity is in the bulk

• Solving hierachy problem: The gravity profile is ‘warped down’ at IR brane where it overlaps very little with SM fields

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5D metric :

ds2 = e−2kRφη μν dx μ dyν − R2dφ2

Warped XD AdS/CFT duality RS bulk is a part of space (a spacetime with a negative

cosmological constant) AdS/CFT duality: The gravity theory in 5D AdS space is

physically equal to 4D comformal gauge field theory The most intriguing feature of RS model

Experimental predictions Gravitons with TeV scale mass and TeV sccoupling to

SM fields Deviation from 4D energy conservation Little constraints from astronomy and cosmology

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AdS5

Conclusions

Extra dimension is a possible and worthwhile extension of 4D physics

Extra dimension is not a new idea, but the structures and mechnisms are variant from model to model so we still need to find out the most possible one

Extra dimension theories have rich phenomena which can be tested on colliders and astronomical observations

Modern extra dimension models may provide us a ladder from low energy world to string theory or other unknow mother theory