Extra DimensionsExtra Dimensionsof Space-Timeof Space-Time

String theory suffers conformal anomaly that makes theory inconsistent --> get rid of it

Conformal anomaly ~ (D-26) for a bosonic string ~ (D-10) for a fermionic string

String theory suffers conformal anomaly that makes theory inconsistent --> get rid of it

Conformal anomaly ~ (D-26) for a bosonic string ~ (D-10) for a fermionic string

• Is it possible that we actually leave in D > 4?• Do we have experimental evidence for D=4, D> 4?

• Is it possible that we actually leave in D > 4?• Do we have experimental evidence for D=4, D> 4?

Motivation:

Why don’t we see extra dimensionsWhy don’t we see extra dimensions

Kaluza-Klein ApproachKaluza-Klein Approach

4 4d dE M K

2 ( ) ( ) ( , )M N m nMN mnds G X dX dX g x dx dx x y dy dy

( )

0

( , ) ( ) ( )nn

n

x y x Y y

Pseudo-Euclidean space

Minkowski space

compact space

Metrics

Fields

K-K modes

Eigenfunctions of Laplaceoperator on internal space Kd

2 2 22 2 1 2

2

... dn

n n nm m

R

Radius of the compact space

Masses

Couplings (4 )(4)

( )

d

d

gg

V

( )d

dV R

Multidimensional GravityMultidimensional Gravity

4 (4 )

(4 )

1ˆ ˆ[ ]16

d dE MN

N d

S d X G R GG

4 (4) (0)

(4)

1[ ] modes

16EN

S d x g R g non zero KKG

(4) (4 )

1N N d

d

G GV

( 1/ 2)(4)( )Pl NM G

12( )

(4 )( ) dN dM G

dV R

Action

K-K Expansion

Newton constant

Plank Mass

2 2dPl dM V M Reduction formula

Low Scale GravityLow Scale Gravity

1 2(4)( ) , r R N

mmV r G

r

d+1d1 2 2

(4 ) 1 12

( )( ) (2 ) , r R

(

)N d d

mmV r G

r

2/2 2 1

Rd

d d PlPl

MM R M

M M

30/ 171 TeV R 10 cmdM -1 3

7 -1

12 -1

2 0.1 R 10 e

3 10 c R 100 e

6 10 c R 10 Me

d R mm V

d R m V

d R m V

| | /1 1 1(4) 1 2 (4) 1 2

0

( ) nm r n r RN Nr r r

n n

V r G m m e G mm e

Modified Newton potential

10

10

10

Brane WorldBrane WorldCompact Dimensions Non-compact dimensions

Kink soliton

Energy density

brane

SMSM NewNew

D4-braneD4-brane

Bulk

Localization on the brane

R

(Potential well)

Space-time of Type I superstring

The ADD ModelThe ADD Model1 / 2

2 ˆ ( , ) MN MN MNdG h x y

M

( )4 4int

1ˆˆ ˆˆ ( , ) ( )nd MNMN

n Pl

S d x GT h x y d x T h xM

/ 21 1

1 10 0

2( )

( 1)

ER dERd d d d

d dn

E S n S n dn R Ed

SM

graviton

m-i n /( ) 1ˆ ( , ) ( ) e my Rn

MN MNn d

h x y h xV

metric

K-K gravitons

Interactions with the fields on the brane

The # of KK gravitons with masses nm E M

Emission rate 2 2

1( )

d

dPl

EE

M M

Particle content of ADD modelParticle content of ADD model

4-dimensional picture• 1 massless graviton (spin 2) + matter• KK tower of massive gravitons (spin 2)• (d-1) KK spin 1 decoupling fields• KK tower of real scalar decoupling fields• KK tower of scalar fields (zero mode – radion)

(0)G( )nG

2( 2) / 2d d ( 2)d

(4+d)-dimensional picture:• (4+d)-dimensional massless graviton + matter

The SM fields are localized on the brane, while gravitons propagate in the bulk

The “gravitational” coupling is 1 / 21/ dM

HEP PhenomenologyHEP PhenomenologyNew phenomena: graviton emission & virtual graviton exchange

• KK states production

221

1 2 2

1dPl md d d

M ddS m

dtdm M dt M

( )ne e G ( )e e

bg

LHC5 TeVM

HEP Phenomenology IIHEP Phenomenology II• Virtual graviton exchange

( ) ( , )ne e G f f HH gg

2 2 2

1 3( 1)

2nPl n n

T TP P dA T T

M s m d s m

( )nG

Spin=2

Angular distribution

SM

2 2

2 11 1

12 2 20

1Pl Pl

dPl

d dM Mn n

M m dmS S

s m M s m

[( 1) / 2]/ 2 1 1 21

4 2 21

( ) ( ) ( )2

dd k d kd

kk

S s si c

M M M M

q

q-1.5 TeVM 0.5 TeVs

Randall-Sandrum ModelsRandall-Sandrum Models

PlankPlankTeVTeV

D4-brane D4-brane

Bulk

15 4 2/E M S Z

0y y y

4 3 (5)ˆ ˆ{2 [ ] }R

MN

R

S d x dy G M R G

1 2

4 (1) 4 (2)1 1 2 2( ) ( )

B B

d x g L d x g L

Metric2 2 ( ) 2yds e dx dx dy

warp factor

Positive tension

Negative tension

Matter( ) | |y k y

3 3 21 2 24 , 24M k M k

Perturbed Metric 2 2 ( ) 2( ( , )) (1 ( ))yds e h x y dx dx x dy

graviton radion

Randall-Sandrum Model cont’dRandall-Sandrum Model cont’d

32 2( 1)k RPl

MM e

k

HierarchyProblem !

PlM

1 TeVk RPlM e

Brane 1

• Massless graviton• massive K-K gravitons

• massless radion

kRn nm ke

Brane 22 ( )Re Wrap factor

2

4 (0) ( )

1

1 1 1

2 3nn

eff BnPl

S d z h T h T TM

• Massless graviton• massive K-K gravitons• massless radion

n nm k

HEP PhenomenologyHEP PhenomenologyThe first KK graviton mode M ~ 1 TeV

• Drell-Yan process • Excess in dijet process

(1) (1)

(1)

, gg

,gg ,gg

qq G l l G l l

qq G qq

Tevatron LHC

Exclusion plots for resonance production

Excluded Excluded

D-Y

Dj

Run I

Run II D-Y

110 fb

1100 fb

( / ) k RPlk M e

HEP Phenomenology IIHEP Phenomenology IIThe x-section of D-Y production

Tevatron (M ~ 700 GeV) LHC (M ~ 1500 GeV)

First KK mode First and subsequent KK modes

( / ) k RPlk M e

0.1 1 0.1 1

HEP Phenomenology IIIHEP Phenomenology III(1)pp G e e

2

(1) 4

(1) 2 4

0 f 1, spin 1 f 1 cos

, f 1 cos

gg , f 1 3cos co

4 s

spin

qq G l l

G l l

LHC

Angular dependence

LHC

ED ConclusionED ConclusionADD Model• The MEW/MPL hierarchy is replaced by• For M small enough it can be checked at modern and future colliders• For d=2 cosmological bounds on M are high (> 100 TeV), but for d>2 are mild• Preictions of modification of the Newton’s law may be checked

2/130/10

d

d

Pl

R M

M M

RS Model• The MEW/MPL hierarchy is solved without new hierarchy• A large part of parameter space will be studied in future collider experiments• With the mechanism of radion stabilization the model is viable• Cosmological scenarios are consistent (except the cosmological constant problem)

What comes beyond What comes beyond the Standard Model ?the Standard Model ?