Physics of Extra Dimensions

Sreerup RaychaudhuriIIT Kanpur

x

y

z

We are used to the idea of three space dimensions ─ where is the room for more dimensions?

Relativity introduces a ‘fourth dimension’, viz. 0x ct

xo

x

Minkowski space

2 22 0ds dx dx

x0 is not really an extra dimension…

'x x

Compact dimensions:

A long human hair looks one-dimensional to us

It still looks one-dimensional to an ant walking along it

It looks two-dimensional to a louse living on it

What determines the number of dimensions is the length scale at which we are doing the experiment

Compact dimensions are those where we must impose periodic boundary conditions…

Typical length scale , e.g. circumference of the cylindrical hair.

0

The compact dimension will not show up in those experiments where the measurements are made at a length scale

0r

ergo, if we choose smaller than the smallest distance experimentally accessible, we can have as many extra spatial dimensions as we wish.

0

Pluritas non est ponenda sine neccessite

William of Ockham c. 1320

We should not introduce extra spatial dimensions unless we actually need them to explain empirical facts…

Compact dimensions were introduced in the early days of quantum mechanics

X=0 X=L

2r = L

Periodic boundary conditions are needed to define a free particle, a Bloch state, etc., etc.….

Most of solid state physics is done on a 3-torus!

History of Extra Space Dimensions

C.H. Hinton 1884 : ‘tesseract’

D.Nørdstrom 1914 : unification of Newtonian gravity with Maxwell equations

T. Kaluza 1921 : unification of Einstein gravity with Maxwell equations

O. Klein 1926 : better version of Kaluza’s theory

W. Pauli 1937 : 6-dimensional Kaluza-Klein theory

……… (limited success)

String theory : bosonic string ‘lives’ only in 26 1970’s

dimensions…

many variations & developments

Revived in 1998 as a solution to the gauge hierarchy problem

SM is an interacting quantum field theory makes no sense as a classical field theory because the particulate nature of quarks, leptons and gauge bosons is well established.

Tree-level calculations correspond only to lowest order term in perturbation expansion make no sense unless ALL the terms in the expansion are considered, at least in principle

2 3(1) (2) (3) ...fi fi fi fi fiS i T i T i T

• Radiative corrections to elementary fermion

masses grow logarithmically as cutoff scale, i.e. Log ─ power-law dependence cancels due to chiral

symmetry remain small

• Radiative corrections to elementary gauge boson masses grow logarithmically, i.e Log ─ power-law

dependence cancels due to gauge symmetry remain small

• Radiative corrections to elementary scalar masses grow quadratically as cutoff scale, i.e. 2 ─ not

protected by any (known) symmetry could become very large

Higher order terms (radiative corrections) can be neglected if and only if they are small

2 3(1) (2) (3) ...fi fi fi fi fiS i T i T i T …

pointed out by (1972)

rule for the self-interactions of the boson

leads to a 2 divergent self-energy correction to the mass

H

H

H H

H

H

H2

22

2~ log finiteHM

Lint = H4

i

22

22

~ log finiteHM

…would drive Higgs mass MH to the cutoff scale

e.g. W+W-H coupling would become non-perturbative !!2 2

2 2H

W W

M

M M

g g

There are two ways out of the hierarchy problem:

1. Postulate a symmetry which will cause the 2 term to cancel ─ supersymmetry, little Higgs models

2. Reduce the cutoff to the TeV scale ─ technicolour, extra dimensions

Energy Scale Cutoff for the Standard Model :

Inputs to the Standard Model:

1. Quark model

2. Electroweak gauge theory : scale ~ 100 GeV

3. QCD : scale ~ 1/3 GeV

i.e. it is known to be valid to ~100 GeV 10-16 cm

Things lacking in Standard Model :

1. Objects more elementary than quarks/leptons ?

2. Grand unification ?

3. Role of gravity

Any of these could provide the reason for a cutoff scale

Natural scale for a quantum theory of gravity : Planck mass

We do not have any compelling empirical reason to believe that quarks/leptons have substructure

We do not have any compelling empirical reason to believe in grand unification

BUT

We do know that gravity exists and that it must be quantized

Ockam’s razor again…

16 21.22 10 TeV /PN

cM c

G

This is so large because gravity is so weak…

345.7 10N

F

G

G

Definite cutoff for SM !

Hierarchy problem: If quantum gravity gives the cutoff for the Standard Model (desert scenario), then the Higgs boson mass will be driven to the Planck scale…

Q. Why is the Planck scale so large?

alternatively:

Q. Why is gravity so weak compared to the other interactions?

100 000 000 000 000 000H P WM M M

Naturalness :

Very large or very small numbers are unstable under quantum corrections

Need some underlying symmetry to protect them

WISHFUL THINKING

If gravity were not so weak, e.g. if GN ~ GF the Standard

Model would be cut off at a ‘Planck scale’ of ~ 100 GeV ─ there would be no hierarchy problem

Can such an idea be a serious scientific possibility?

We have measured the strength of the gravitational field many many times, since the days of Isaac Newton… even in high school labs... today there is no doubt at all that GN is indeed small…

BUT

The length scales at which such measurements have been done are very large compared to atomic sizes…

Could it be that gravity is weak at large scales, but strong at small scales…. ? i.e. smaller than the electroweak scale: 10-16 GeV

Then the much lower energy scale of this strong short-range (quantum) gravity would automatically cut off the Standard Model at much lower energies

Known: We cannot achieve this within the framework of Einstein gravity in (1+3) dimensions

1

28 NR Rg G T

, 0,1,2,3

Is the talk over ?

It can be done if there are extra compact dimensions

NEWS FLASH

Roughly speaking, there are two main classes of extra-dimensional models for making gravity strong at small length scales :

1. Gravitational lines of force are dispersed in the extra dimensions and only a small number are observed in four-dimensional experiments : force is weakened in proportion ─ Arkani-Hamed, Dimopoulos and Dvali 1998

2. Gravity is strong in some other region of space, and loses strength as it ‘shines’ on our four-dimensional space : force is weakened according to distance ─ Randall and Sundrum 1999

Both paradigms work if and only if there is a mechanism to confine the experiment(er) within the four Minkowski dimensions

i.e. the extra dimensions are ‘seen’ by gravity alone

What do we know experimentally about the length scale to which Einstein gravity (effectively Newton gravity, or just the inverse square law) is valid?

On astronomical scales, inverse square law is valid

Kepler (1619)… Hooke (1660 ?)… Newton (1687)

/( ) 1Nm

rG mr e

r

Dark matter discovery...

TASI 2004

Cavendish 1798

Eötvös 1891

torsion balance

Torsion balance experiments at length scales ~ few cm

B. Heckel

Extremely sensitive torsion pendulum : tungsten torsion fibre 20 m thick

Rotating disk with holes ─ matching holes in pendulum

torsion effect cancels finely for inverse square law

any deflections of laser beam will be due to deviations from inverse square law

Eöt-Wash experiment at length scales ~ 100 m

E.Adelberger

/( ) 1Nm

rG mr e

r

For ||~1

< 150 m

2003 data

Eventually

< 60 m

Compare with29~ 10 mP

Other interactions ─ electroweak, strong ─ have been tested all the way down to the electroweak length scale

12 16~ 10 m = 10 cmEW

Einstein gravity in (1+3) dimensions has been tested only up to the scale of

2 2~ 10 m = 10 cmEotWash

Can there be extra dimensions a bit smaller than this, e.g. 10-3 cm ?

Many electroweak precision tests would show up new effects if there were extra dimensions in which the carrier fields could propagate… but they do not show any such effects…

We require that only gravity should ‘see’ extra dimensions … other interactions should not !

Gravity

SM fields

ADD Model : Large Extra Dimensions

Arkani-Hamed, Dimopoulos and Dvali (March 1998)

‘d’ compact dimensions

1 +3•1+3 Minkowski dimensions

• ‘d’ large compact dimensions

• SM fields trapped in 1+3

• Gravity propagates in 1+3+d10-3 cm

Mechanism of confinement? …. Domain walls… Vortices…. D-branes….

D-branes:

• Introduced by Polchinksi in 1995• Solitonic configurations of superstring theory

• Dp brane is a 1 + p dimensional hypersurface

• open strings have ends fixed to Dp branes

Dirichlet boundary conditions

Fields which are stringy excitations are confined

within length 1// (/ = string tension)• Closed strings are free to propagate in 10 dimensions

String theory provides the ideal mechanism to confine SM fields in 1+3 dimensions

ADD Model : String Theoretic View

10-3 cm

Antoniadis, Arkani-Hamed, Dimopoulos and Dvali (April 1998)

10-17 cm

•Observable Universe is a D3 brane•Max. no of extra dimensions is d = 6

•SM fields: spin 0, ½ and 1 excitations of open strings with ends confined to D3 brane

•Gravitons: spin 2 excitations of closed strings propagating in bulk

String tension can be as small as -1 ~ 1 TeV stringy excitations at a TeV

D 3 bra

ne

bulk

Weak gravity

Qualitative :

Quantitative : Einstein-Hilbert action in 4+d dimensions

4

4

1( , )

16

(1

ˆ

)

ˆˆ

6

ˆdB

B

B

N

d x xS d y g yG

Vd x g x

G

BR

R +...

Lines of force are mostly dissipated in the bulk…

Only a small number are intercepted by the brane

ˆN

Nd

GG

V

Integrate over bulk for large objects

Bulk scale versus Planck scale

2

2

2

2 P

d

d

dCR

M

on a d-torus

Possible to have TeV strings if d 2

2

2

2

2

2 2 ; ;

ˆ ˆˆ

ˆ

N N BdP

d

NP

B

N

PP

G G VM

M

G GM

V

M

ADD Solution to the Hierarchy problem:

1. All known experiments/observations are done on the D3 brane and do not sense the extra dimensions until the energy scale of the experiment reaches the bulk scale (string tension)-1 (= TeV?)

2. Gravity propagates in all the 3+d spatial dimensions, including the D3 brane, of course.

3. As we approach the bulk scale, stringy excitations begin to appear, i.e. the Standard Model is no longer valid

4. Bulk scale (= TeV?) acts as a cutoff for the Standard Model

5. There is no hierarchy problem…

Observable consequences :

2 22

2 2

2 2. / 22

2 20

2 2. / 22

220

ˆˆ ( , , ) 0

ˆ ( , , ) 0

( , ) 0

( , ) 02

C

C

t

in y Rt n

n

in y Rt n

n C

t x y

t x yx y

t x ex y

nt x e

x R

2

1,2 ( , ) 0 ( ..., )2

n d

C

nt x n n n

R

Massless bulk scalar

Fourier series on a d-torus

Massive scalars on brane

2 2 221 2

2

...

22

dn

CC

n n nnM

RR

Tower of Kaluza-Klein states : ( )n x

Spacing between states :

1~

~ if ~

~ 0.01 eV if ~ 0.001 cm

nC

P C P

C

MR

M R

R

On the brane…

No of contributing states :

13100 GeV~ ~ 10

0.01 eVn

s

M A bulk scalar field is like a huge swarm of 4-scalar fields on the brane

Position of the brane is at 0y

Standard Model fields live only on the brane :

ˆ ,x x y y

Interaction with single bulk scalar field is the same as interaction with a swarm of 4-scalar fields on the brane

4int

4 .

4

/ 2

4

ˆ ˆ ˆ( , ) ( , ) ( , )

ˆ ˆ ˆ( , ) ( , ) ( )

ˆ ˆ

0 0

0 0

( ) ( ) ( )

ˆ ˆ( ) ( ) ( )

Cn

n

n

d

in y R

n

n

d

n

S d x x x x

d x x

d y y y

x x

d x x x x

d x x x

d y y e

x

Weak gravitational field limit : ( ) ( )g x h x 16

PM

Valid for energies much lower than Planck (bulk) scale

ˆˆ (ˆˆ , ) 0xh y

1ˆ ˆˆ ˆ2

ˆ ˆ ˆ 0R Rg Free Einstein equations in 4+d dimensions :

reduce to :

Massless Klein-Gordon equation for a bulk tensor…

ˆˆ

( , ) ( , )ˆ ( , )( , ) ( , )

i

j ij

h x y A x yh x y

A x y x y

Each of the fields has its own Kaluza-Klein decomposition

( , ), ( , ), ( , )i ijh x y A x y x y

On the brane…

All the bilinear covariants with Standard Model fields have indices running over 0,1,2,3 only

ˆˆi t ˆ

4n

4

ˆˆ ˆ( , ) ( , ) ( , )

28

( ) ( ) (

0 0

)

d

n

n P

S d x x x x

d x x x h x

d y h y

M

y

Interaction with a graviton tower

4int

4

ˆ ˆ( , ) ( , ) ( , )2

8( ) ( ) (

0

)

0d ijij

n

n P

d y y yS d x x x x

d x x x xM

Interaction with a dilaton tower( , ) ( , )iii

x y x y

Han, Lykken and Zhang, Phys Rev D59, 105006

Feynman Rules for the ADD model

all scalars

all gauge bosons

all fermions

4int ( )2

nSM

n

S d x h T

Collider physics with gravitons/dilatons:

• Graviton tower couples to every particle-antiparticle pair

• Blind to all quantum numbers except energy-momentum

• Each Kaluza-Klein mode couples equally, with strength

• Tower of Kaluza-Klein modes builds up collectively to an observable effect

• Individual graviton modes escape detection missing

• Signals will show

1. excess over Standard Model cross-sections

2. different distributions due to spin-2 nature

3. energy and momentum imbalance

Tp

REAL GRAVITONS

n 2

n

Incoherent sum nn

VIRTUAL GRAVITONS

n

2

n

Coherent sum nn

A A

Sum over KK states can be done using the quasi-continuum approach

0

2

/ 2

( ) ( ) ( )

( )4 ( / 2)

s

nn

d dCd

A M dM M A M

R MM

d

2

2 44

( , ) 1ˆ

n n SP

d s

s M i MM

Sum over propagators…

reduces to a contact interaction…

Important processes at colliders

LHC

ILC

* , , , ,npp G WW ZZ JJ

, , ,n n n npp G WG ZG JG

, , ,n n n npp G WWG ZZG JJG

, ,n n ne e G ZG JG * , , , ,ne e G WW ZZ JJ

, , ,n n n ne e G WWG ZZG JJG

0.87 0.720.65

1.091.45

0.380.6

30

4

1

0.1

1

10

100

d=2 d=3 d=4 d=5 d=6

Bounds on bulk scale ‘string’ scale

(TeV)SM

Black : LEP & Tevatron Run II

Green : SN 1987A

1. Find out of there are signals for KK towers of gravitons ─ large-pT excess, missing energy, etc.

2. Determine whether the signals are indeed due to brane-world gravitons and not some other new physics ─ gravitons would be blind to all Standard Model quantum numbers

3. Identify these particles (if seen) as graviton modes ─ spin-2 nature is a dead giveaway

4. Find out the number of large extra dimensions

5. Find out the radius of compactification RC, or equivalently, the bulk scale (string scale MS)

6. Find out the geometry of the extra dimensions

7. Find out dynamics which makes some dimensions large & some small

Important issues in ADD phenomenology

Dutta, Konar, Mukhopadhyaya, SR (2003)

1 TeV 2 TeV

Laboratory Black holes

Gravity becomes strong at ~ TeV.

LHC will collide protons at 14 TeV Trans-Planckian regime

Schwarzschild radius for a black hole in 3+d dimensions:

In a pp collision, if the impact parameter is less than RS the protons will coalesce to form a micro-black hole.

Cross section is just: BH RS2 (semi-classical)

For ~ 1 TeV there will be copious black hole production

Decay by Hawking radiation: produces distinctive signatures

1

12

8 11ˆ 2ˆ

d d

S

PP

mR

dMM

ˆPM

‘CATFISH’ generator… 31.08.2006

Simulation of a black hole production and decay event at the LHC (de Roeck 2003)

• The KK modes have masses typically : 10-3 eV• The scale of strong gravity is typically : 1012 eV

– scale hierarchy of 15 orders of magnitude

• Quantum corrections tend to shrink the size of the d-dimensional bulk– process terminates only when the scale reaches Planck

scale back to ‘tHooft 1972…

• Large extra dimensions are unstable ! – Need a mechanism to stabilize the size…

The Hierarchy problem strikes back…

All is not well with the ADD model…

Randall-Sundrum Model

warped compactification

May 1999

Model is based on an orbifold compactification……one extra dimension…

0

2 CR

1 2/S ZA circle folded about a diameter

Fixed points

Only logical place to place a brane is at a fixed point ─ put one at each

Matter content of a brane is parametrized as a VEV ─ brane tension cosmological constant for 4-D Einstein gravity on the brane

41 ˆˆ( , ) ( , )ˆ16

ˆRS C

N

S d x d R g x xG

- R

5-D Einstein-Hilbert action with a cosmological constant term :

Different normalization convention:3

1ˆ ˆ ˆ ˆ ˆ32 16

C CN

P N

R RG

M G

4 3ˆ ˆˆ ( , ) 2 ( , )ˆRS C PS d x d R g x M x - R

40ˆ ( ,0) ( ,0)d x g x V x 0- L

4 ˆ ( , ) ( , )d x g x V x - L

Equations of motion:

1ˆ ˆˆ ˆ2

(0) (0) ( ) ( )ˆ ˆ ˆˆ ˆ ˆ03

ˆ ˆˆ ˆ

1 ˆˆ ˆˆ

P

g R Rg

g g g g V g g VM

‘RS Ansatz’ : 22 2 ( )

2 ( ) 0ˆ

0 1

fC

f

ds e dx dx d R

eg

Cf kR Solution:0

3 3 3

ˆ

ˆ ˆ ˆ24 24 24P P P

V Vk

M M M

Fine-tuning :

RS Metric : 22 | |2 CkRCds e dx dx d R

Warp : metric dies out exponentially from = 0 to =

AdS5

Metric contracts exponentially along the ‘AdS5 throat’ measuring sticks contract exponentially wavelengths increase exponentially energies drop exponentially

Like a gravitational redshift

RS Mechanism:

At = 0 :

At = :

1CkRe

16~ 10 if 12C CkR kRCe e kR

Weakness of gravity on ‘TeV brane’ at = is explained without recourse to large numbers

Randall-Sundrum solution to the hierarchy problem

All mass scales on the ‘Planck brane’ get scaled by warp factor when they get ‘shined’ on the ‘TeV brane’

CkRe

1/32

1/32 12ˆ PP P

C

MM kM

R

If we set then

is large no TeV strings

~C PR ˆPM

Kaluza-Klein modes of the RS bulk graviton field :

Small fluctuations around vacuum metric

2 ( )( ) CkRe xx hg

Fourier expansion of graviton field :

0

(1

( , ( )) ) nn

nC

h x h xR

Warped harmonics

4 222

1( ) ( )C CkR kR

n n nC

d de M e

R d d

2 ( ) ( )CkRm n mnd e

Equation of motion :

2 ( ) 0nnhM x

Goldberger and Wise (1999) Davoudiasl, Hewett and Rizzo

Conformal coordinates :2 2( ) ( ) ( )C CkR kR

n nn

nz e f ek

M

Eigenvalue equation : 2

2 22

4 0n nn n n n

n n

d f dfz z z f

dz dz

Bessel equation of order 2

2

2( ) ( ) ( )CkR

n nnn

n n

eJ z Y

Nz

Require harmonics to be continuous at the orbifold fixed points…

0

0

22

2

1( )

C

C

C

kRn n n

kRn n

n nm R

M x ke x m

x e

N J x

Electroweak scale

1( ) 0nJ x

Warped harmonics :

Graviton interaction with matter :

0int

1

8 8( ) ( ) ( ) ( ) ( )

CkRn

nP P

ex h x T x h x T x

M M

L

Zero (massless) mode gives usual Einstein gravity

Massive (attenuated) modes have electroweak strength couplings

RS Gravitons are like WIMPs : masses and couplings both resemble electroweak particles

Two free parameters: 1 1 0M x m

8

P

k

M

Feynman rules same as in ADD model apart from warp-up factor…

• RS graviton width grows rapidly with graviton mass– Only first three modes can form narrow resonances– For large part of parameter space only first resonance is viable

• RS gravitons decay to all particle pairs• Maximum BR is to jets; sizeable width to WW and ZZ

• No deviations from SM at LEP-2

lightest RS graviton is heavier than 210 GeV

• Tevatron Drell-Yan data show no deviations either

lightest RS graviton is heavier than ~ 850 GeV

• LC: smaller but clean final states: • graviton resonances in Bhabha scattering and e+e- +-

RS graviton phenomenology :

2 40 0

( )( )

CkRs

n n n n

xeD s

s M iM m

0s

sx

m

e e Graviton resonances:

Davoudiasl, Hewett and Rizzo

Modulus stabilization and the radion:

16~ 10 if 12CkRCe kR

Warping is extremely sensitive to RC

( ) 222 2(( ))T xk T xds e g x d

Consider the radius of the extra dimension as a dynamical object :

( )T xModulus field :

3ˆ2 ( )4( ) kM

k

T xex Radion field :

3ˆ1 1 24 22 12

( ) Mgrav k

S d x g x

Radion is a free field i.e. can assume any value same for modulus

Need for modulus stabilization

Goldberger-Wise mechanism :

Assume a bulk scalar field ( , )B x

Write down a B4 theory in the bulk and on the two branes…

Solving the equation of motion for (x) and integrating over leads to potential with a steep minimum at

20

2

v4( ) log

vCB

kkR k T x

M

Can assume the desired value without assuming any large/small numbers…

Undetermined parameters: radion mass & radion vev

M

Radion couplings are very Higgs-like…

Radion phenomenology :

• Radion phenomenology is rather similar to Higgs phenomenology for tree-level processes

• Possibility of Higgs-radion mixing– Giudice, Rattazzi, Wells (2000)– Huitu, Datta (2002)

• ‘Radion’strahlung process …– Production is just like Higgs-strahlung– At one-loop, effect of kinetic terms in radion-fermion couplings

becomes important – Try to identify radion by its somewhat different decay widths to

gluons (one-loop) i.e. dijet decay mode

Light radion decays primarily to gluon jets; Higgs decays primarily to b-jets

Use b-tagging to compare cross-sections for

ande e ZJJ e e Zbb

Ratio shows distinct difference…

Das, Rai, SR, PLB 2005

Main Issues in RS Phenomenology :

• Find out of there are signals for graviton resonances ─ bump hunting…

• Determine whether the resonances are indeed RS gravitons and not some other new physics ─ RS graviton masses are spaced like zeros of Bessel function J1

• Identify these resonances as graviton modes ─ spin-2 nature is a dead giveaway

• Find out if there are signals for radions─ very similar to Higgs search

• Find out the mass and coupling parameters ─ mass and width measurements (like W,Z at Tevatron)

• If the resonances are broad distinguish between RS and ADD models

• Distinguish the radion from a Higgs scalar

Universal Extra Dimensions :

Both ADD and RS models give effective field theories below the cutoff scale ~ TeV

There will be higher-dimension operators, suppressed by electroweak scale mass only can cause all sorts of trouble for SM, including proton decay

Place all the SM fields in the bulk and do away with the brane

Orbifold the bulk to get chiral fermions; no branes

Expand all fields in terms of the bulk harmonics each SM field has its own Kaluza-Klein tower.

KK number is conserved due to orthonormality of bulk harmonics non-diagonal operators are suppressed by RC 1

ˆPM

Appelquist, Cheng, Dobrescu 2002

Dimensional Deconstruction :

Many people are uncomfortable with the idea of extra space dimensions

Gauge theory based on complicated gauge group, with different sets of fermions, each transforming as different representations of the gauge groups

At low energies, it resembles a 5-D theory

At still lower energies, it resembles a tower of KK states…

Arkani-Hamed, Cohen, Georgi 2001

Holography :Maldacena conjecture 1997

Can the AdS5 bulk be the dual of some 4-dimensional CFT on the branes?

AdS/CFT correspondence

Presence of branes corresponds to s.s.b. of conformal invariance, possible in an effective theory…

Only reality is 4-D Standard Model and 4-D CFT at some high scale ; extra dimensional is a duality artefact…

Recommended reading :

Field-theoretic aspects…

AdS/CFT, model building…