Download Compact dimensions

Physics of Extra Dimensions
Sreerup Raychaudhuri
IIT Kanpur
z
y
We are used to the idea of
three space dimensions ─
where is the room for
more dimensions?
x
Relativity introduces a ‘fourth dimension’, viz.

x 0  ct
 

x'   x
xo
Minkowski space
x
    dx 
ds  dx
2
0
2
2
x0 is not really an extra dimension…
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
cylindrical hair.
0,
e.g. circumference of the
The compact dimension will not show up in those
experiments where the measurements are made
at a length scale
r
0
ergo, if we choose 0 smaller than the smallest
distance experimentally accessible, we can have as
many extra spatial dimensions as we wish.
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
2r = L
X=0
X=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
………
String theory
1970’s
(limited success)
:
bosonic string ‘lives’ only in 26
dimensions…
many variations &
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
S fi   fi  iT
(1)
fi
  i  T
2
(2)
fi
  i  T
3
(3)
fi
 ...
Higher order terms (radiative corrections) can
be neglected if and only if they are small
S fi   fi  iT
  i  T
(1) …
fi
2
(2)
fi
  i  T fi(3)  ...
3
• 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
rule for the self-interactions of the
boson
H
H
Lint = H4
H
H
leads to a 2 divergent self-energy correction to the mass
H
 M ~    log
2
H
H
2

2

2
H
i
pointed out by
(1972)
 finite
 M ~    log
2
H
2

2

2
 finite
…would drive Higgs mass MH to the cutoff scale 
 e.g. W+W-H coupling would become non-perturbative !!
g M H2
2 MW

g 2
2 MW
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 
Ockam’s razor again…
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
Natural scale for a quantum theory of gravity : Planck mass
c
MP 
GN
1.22 1016 TeV / c 2
This is so large because gravity is
so weak…
Definite cutoff for SM !
GN
GF
5.7 1034
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…
M H  M P  100 000 000 000 000 000 MW
Q. Why is the Planck scale so large?
alternatively:
Q. Why is gravity so weak compared to the other interactions?
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
R  Rg  8 GNT
1
2
 ,  0,1, 2, 3
Is the talk over ?
NEWS FLASH
It can be done if there are extra compact dimensions
Roughly speaking, there are two main classes of extradimensional 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)

GN m
m (r ) 
1   e r / 
r

Dark matter
discovery...
TASI 2004
Torsion balance experiments at length scales ~ few cm
Cavendish 1798
Eötvös 1891
torsion balance
Eöt-Wash experiment at length scales ~ 100 m
E.Adelberger
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

GN m
r / 
m (r ) 
1e
r
2003
data

For ||~1
 < 150 m
Eventually
 < 60 m
Compare with
P
~ 1029  m
Einstein gravity in (1+3) dimensions has been
tested only up to the scale of
~ 10  m = 10 cm
2
2
EotWash
Can there be extra dimensions a bit smaller than
this, e.g. 10-3 cm ?
Other interactions
─ electroweak, strong ─
have been tested all the way
down to the electroweak length
scale
EW
~ 10
12
 m = 10 cm
16
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 !
SM fields
Gravity
ADD Model : Large Extra Dimensions
Arkani-Hamed, Dimopoulos and Dvali (March 1998)
‘d’ compact
dimensions
•1+3 Minkowski dimensions
• ‘d’ large compact dimensions
• SM fields trapped in 1+3
• Gravity propagates in 1+3+d
10-3 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
Antoniadis, Arkani-Hamed, Dimopoulos and Dvali (April 1998)
•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
bulk
10-17 cm
10-3 cm
String tension can be as small as -1 ~ 1 TeV  stringy excitations at a TeV
Weak gravity
Qualitative :
Lines of force are
mostly dissipated in
the bulk…
Only a small number
are intercepted by
the brane
Quantitative : Einstein-Hilbert action in 4+d dimensions
Sˆ 
1
4
d
d
x
d
y  gˆ B ( x, y ) Rˆ B

16 Gˆ B
VB
4

d
x  g ( x) R + ...

16 GN
Integrate over bulk
for large objects
Gˆ N
 GN 
Vd
Bulk scale versus Planck scale
2
GN  2 ;
MP
Gˆ N 
2
;
2 d
ˆ
MP
Gˆ N  GNVB
 MP 
 Mˆ P  

 V 
 B

MP 
2
2 d
2
2 d
 2 RC 
d
2 d
on a d-torus
Possible to have
TeV strings if d  2
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 :
ˆ (t , x , y )  0
ˆ
Massless bulk scalar
 2 2
2  ˆ
   t  2  2   (t , x , y )  0
x
y 

 2 2
2  
   t  2  2    n (t , x ) ein . y / 2 RC  0
x
y  n 0

Fourier series
on a d-torus
 2 2

n2
in . y / 2 RC

   t  2 

(
t
,
x
)
e
0
n
2

x
2

R
n 0 


C





n2
 
  n (t , x )  0 n  (n1, ..., nd )
2


2

R


C


Massive scalars on brane
On the brane…
Tower of Kaluza-Klein states :  n ( x)
n2
Mn 
 2 RC 
2
n12  n22  ...  nd2

2 RC
Spacing between states :
M n ~
1
RC
~ MP
if RC ~
P
~ 0.01 eV if RC ~ 0.001 cm
No of contributing states :
s
100 GeV
~
~ 1013
M n 0.01 eV
A bulk scalar field is like a huge
swarm of 4-scalar fields on the brane
Position of the brane is at y  0
Standard Model fields live only on the brane :
ˆ  x, y    y 
  x  
ˆ ( x, 0)
ˆ ( x, 0)  y  
ˆ ( x, y )
Sint   d 4 x d d y  
ˆ ( x, 0)
ˆ ( x, 0)  y   
ˆ n ( x) ein . y / 2 RC
  d 4x d d y  
n
ˆ ( x)  
ˆ n ( x)
  d 4 x   ( x)
n
ˆ ( x) 
ˆ n ( x)
   d 4 x   ( x)
n
Interaction with single bulk scalar field is the same as
interaction with a swarm of 4-scalar fields on the brane
Weak gravitational field limit : g  ( x)      h ( x)
16
Valid for energies much lower than Planck (bulk) scale  
MP
Free Einstein equations in 4+d dimensions :
reduce to :
1 ˆ
Rˆ 

Rgˆ 
ˆˆ
ˆˆ 0
2
ˆ hˆ
ˆ ˆ ( x, y )  0
Massless Klein-Gordon equation for a bulk tensor…
h ( x, y ) Ai ( x, y ) 

hˆ
ˆ ˆ ( x, y )  

A
(
x
,
y
)

(
x
,
y
)
ij
 j

Each of the h ( x, y ), Ai ( x, y ),  ij ( x, y ) fields has its own KaluzaKlein decomposition
On the brane…
All the bilinear covariants with Standard Model fields have indices
running over 0,1,2,3 only
 ˆ
ˆ ( x, 0)  y   ˆ  ˆ h ˆ ˆ ( x, y )
Sint   d x d y  ( x, 0)  
  
4
d
2
8
4
 d x
 ( x)   ( x)hn ( x)
MP
n
Interaction with a graviton tower
 ˆ
ˆ ( x, 0)  y   ij  ( x, y)
Sint   d x d y  ( x, 0)
ij
4
d
2
8
4
 d x
 ( x)   ( x) n ( x)
MP
n
 ( x, y)   ii ( x, y)
i
Interaction with a dilaton tower
Feynman Rules for the ADD model
Sint    d 4 x
n
Han, Lykken and Zhang, Phys Rev D59, 105006
all scalars
all gauge bosons
all fermions

n

h
T
2  ( SM )
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 pT
•
Signals will show
1. excess over Standard Model cross-sections
2. different distributions due to spin-2 nature
3. energy and momentum imbalance
REAL GRAVITONS
n

n
Incoherent sum
   n
n
2
VIRTUAL GRAVITONS

2
n
n
Coherent sum A 
A
n
n
Sum over KK states can be done using the quasi-continuum
approach
 A(M
s
n
n
 (M ) 
)   dM  ( M ) A( M )
0
d
C
d /2
R M
 4 
d 2
(d / 2)
Sum over propagators…

sM
n
2
2
n
 i

 (d , s)
Mˆ P4
1
 4
MS
reduces to a contact interaction…
Important processes at colliders
pp   Gn , WGn , ZGn , JGn
LHC
pp 
 
Gn ,WWGn , ZZGn , JJGn
pp  G 
*
n
 
,  ,WW , ZZ , JJ
e e   Gn , ZGn , JGn
 
ILC
 
*
n
 
 
e e G 
e e 
 
,  ,WW , ZZ , JJ
Gn ,WWGn , ZZGn , JJGn
Bounds on bulk scale  ‘string’ scale
100
MS
30
(TeV) 10
1.45
1.09
1
4
0.87
0.72
0.6
1
0.65
0.38
0.1
d=2
d=3
d=4
d=5
Black : LEP & Tevatron Run II
Green : SN 1987A
d=6
Important issues in ADD phenomenology
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
1 TeV
2 TeV
Dutta, Konar, Mukhopadhyaya, SR (2003)
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:

d

1
m 8 1  2

RS 
ˆ
ˆ
M

 MP  P d  2
 
1
d 1


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 Mˆ P ~ 1 TeV there will be copious black hole production
Decay by Hawking radiation: produces distinctive signatures
‘CATFISH’ generator… 31.08.2006
Simulation of a black hole production and decay event at the LHC
(de Roeck 2003)
All is not well with the ADD model…
• 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…
Randall-Sundrum Model
May 1999
warped compactification
Model is based on an orbifold compactification…
…one extra dimension…

 
    2 RC
  

Fixed points
 0
S1 / Z 2
A circle folded about a diameter
Only logical place to place a brane is at a fixed point ─ put one at each
5-D Einstein-Hilbert action with a cosmological constant term :
SRS
1
4
ˆ  Rˆ ( x,  ) 

ˆ

d
x
d

R
g
(
x
,

)


C



16 Gˆ N
Different normalization convention:
RC 1
ˆ
GN 
32 Mˆ P3
ˆ

RC ˆ

ˆ
16 GN
ˆ  2Mˆ 3 Rˆ ( x,  ) 
S RS   d 4 x d  RC  - gˆ ( x,  ) 
P


  d 4 x - gˆ ( x, 0)  V0  L0 ( x, 0) 
  d 4 x - gˆ ( x,  )  V  L ( x,  ) 
Matter content of a brane is parametrized as a VEV ─ brane tension
 cosmological constant for 4-D Einstein gravity on the brane
Equations of motion:
1 ˆ

 gˆ  Rˆ 

Rgˆ 
ˆˆ
ˆˆ
2
1 
(0) (0)  
( ) ( )  

ˆ
  3  gˆ gˆ 



g
g


V


g
g


V
ˆ
ˆ
ˆ
ˆ
 ˆ  0
 ˆ  

Mˆ P 
‘RS Ansatz’ :
ds  e
2
 dx dx  d  RC 
2 f ( )

e2 f ( )
gˆ  
0

Solution:
f    kRC 
RS Metric :
ds  e
2

0

1
2
Fine-tuning :
ˆ
V0
V

k


3
3
ˆ
ˆ
24M P 24M P 24Mˆ P3
 dx dx  d  RC 
2 kRC | |


2
Warp : metric dies out exponentially from  = 0 to  = 
AdS5
RS Mechanism:
Metric contracts exponentially along the ‘AdS5 throat’
 measuring sticks contract exponentially
 wavelengths increase exponentially
 energies drop exponentially
Like a gravitational redshift
 kRC 
1
 kRC 
 e kRC ~ 1016 if kRC  12
At  = 0 :
e
At 
e
= :
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’
Mˆ P
 kM 
2
P
1/ 3
1/ 3
 12M 


R
 C 
2
P
e  kRC 
If we set RC ~
P
then Mˆ P
is large  no TeV strings
Kaluza-Klein modes of the RS bulk graviton field :
Small fluctuations around vacuum metric
g  ( x)  e
2 kRC
    h ( x) 
Fourier expansion of graviton field :
h ( x,  ) 
1
RC
Goldberger and Wise (1999)
Davoudiasl, Hewett and Rizzo

 h ( x) 
n
n 0
n
( )
Equation of motion :


n
 M n2 h
( x)  0
Warped harmonics
1 d  4 kRC d 
2 2 kRC
e

(

)

M
 n ( )
n e

 n
2
RC d 
d 
2 kRC
d

e
 m ( )  n ( )   mn

Conformal coordinates :
Eigenvalue equation :
M n 2 kRC
zn ( ) 
e
f n ( )  e 2 kRC  n ( )
k
2
d
fn
df n
2
2
zn

z

z
n
n  4 fn  0
2
dzn
dzn


Bessel equation of order 2
Warped harmonics :
e2 kRC
 n ( ) 
 J 2 ( zn )   nYn ( zn )
Nn
Require harmonics to be continuous at the orbifold fixed points…

M n  xn ke
 kRC 
 xn m0
 n  xn2 e2 kR 
C
Nn 
1
m0 RC
Electroweak scale
J 2 ( xn )
J1 ( xn )  0
Graviton interaction with matter :
kRC 
8 0
8

e
Lint ( x)  
h ( x) T  ( x) 
MP
MP

n

h
(
x
)
T
(x)
 
n 1
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
Feynman rules same as in ADD model apart from warp-up
factor…
Two free parameters:
M1  x1m0
8 k
MP
RS graviton phenomenology :
• 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-  +-

D( s )  
 e kR 
C
n 0 s  M  iM n  n
2
n

 ( xs )
m04
s
xs 
m0
Graviton resonances:
e e     
Davoudiasl, Hewett and Rizzo
Modulus stabilization and the radion:
Warping is extremely sensitive to RC
e kRC ~ 1016 if kRC  12
Consider the radius of the extra dimension as a dynamical object :
ds  e
2
Modulus field :
Radion field :
S grav
2 kT ( x ) 
g  ( x)  T ( x) d 2
2
T ( x)
( x) 
24 Mˆ 3
k
e
 k T ( x )
1
1
2 Mˆ 3 

2

  d x  g ( x) 2       12   k


4
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
v0
4k 2
kRC  k T ( x) 
log
2
 MB
v
Can assume the desired value without assuming any
large/small numbers…
Undetermined parameters: radion mass M  & radion vev   
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
e e  ZJJ
and
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 :
Appelquist, Cheng, Dobrescu 2002
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
Mˆ P
Dimensional Deconstruction :
Arkani-Hamed, Cohen, Georgi 2001
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…
Holography :
Maldacena conjecture 1997
AdS/CFT correspondence
Can the AdS5 bulk be the dual of some 4-dimensional CFT on the branes?
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…