Transcript Physics of Extra Dimensions A potential discovery for LHC/ILC

Physics of Extra Dimensions
-A potential discovery for LHC/ILC Abdel Pérez-Lorenzana
CINVESTAV
PASI-2006, Puerto Vallarta. México.
Program

Introduction: Why considering Extra Dimensions?

Dimensional reduction: The Effective Field Theory

KK decomposition on torii and orbifolds

General phenomenological aspects

Gravitons Phenomenology

Phenomenology of XD matter fields

KK modes of Matter Fields: Universal Extra Dimensions

New Theoretical ideas for the use of XD
Wonders from the XX century
 Fundamental interactions are described using two different frameworks:
Electroweak and Strong forces in the Standard Model:
Gauge Quantum Field Theories
SU (3) c  SU (2) L  U (1) Y
with matter in irrep
Q (3,2, 31 ); L (1,2,-1); e R (1,1,-2); u R (3,1, 43 ); d R (3,1,- 23 )
Symmetry Breaking by Higgs Mechanism
Q = T3 + 21 Y
H (1,2,1)
Hierarchy problem
(
: dm H2  - l - f
+
2
)L
2
Gauge and Flavor Problems…
Gravity as Geometry of the Space Time: General Relativity
Rmn
- 21 g mn R = 8p G4 N Tmn
c
Not a Quantum Theory (non renormalizable),…. DM, DE, …
The Unification Scheme
Gravity
Theory of
Everything??
Weak
E-M
QG ??
Strong
While MGUT is calculable, the only
plausible scale for Quantum Gravity
seems to be the Planck Scale
5
M Pc
2
=
c 
8p G N
~ 2 . 4  10
18
GeV
The Dream for a Unified Theory
Non Rel QM
QFT
GN  0
ToE[α(GN ,ħ,c)]
c 0
ħ0
Newtonian Mech
The best known candidate for
the ToE is String Theory…
String Theory and Extra Dimensions
String Theory needs extra compact spacelike dimensions to be consistent ( D=10 or
D=11 for M Theory ).
Moduli Space of M Theory
HE
Sugra D=10
S
XD are compact and usually assumed to be
small
T
S
HO
M-Theory
Type II A
S
L Planck
T
Ω
Type I
Type II B
The scale of ST
ST
In the Perturbative Heterotic String Theory Gravity and Gauge
interactions have the same origin, as massless modes of the
closed Heterotic String, and they are Unified at the String Scale
Along the 80’s some authors (Antoniadis,
Benakli, Quirós) suggested the possibility
of having intermediate scales
M S = as M P
a S  0.04
M * ~ v M P ~ 10 11 GeV
Recent developments on String Theory have given support to the idea that (some)
extra dimensions could rather be larger than Planck length (Horava-Witten,´96)
Can MString « MPlanck ?
D-brane models and XD
Type I Sring Theory framework
3-brane
Dp-brane
Brane world.- Our Universe could be
described as a hyper-surface extended in p
spatial dimensions: a p-brane
R >> r >> ℓP
Open String
Closed String
( Gravity )
r
(Gauge)
Bottom-up: To study of these models
we can use an effective field theory
description with M* as the UV-cutoff.
Matter would be trapped to the brane,
whereas gravity propagates on all 4+n
dimensions
In 1998 Arkani-Hammed, Dimopoulus and
D’vali made the key observation that extra
dimensions could even be of millimeter size
and M* as low as few TeV !!
D-brane models and XD
Type I Sring Theory framework
3-brane
Dp-brane
Open String
Closed String
( Gravity )
r
(Gauge)
Experimental signals of TeV scale strings in
LCH/ILC may come from:
• new compactified parallel dimensions (Dbrane SM)
• new extra large transverse dimensions and
low scale quantum gravity
• genuine string and quantum gravity effects
There exist interesting implications in non
accelerator experiments due to bulk states
Bottom-up: To study of these models
we can use an effective field theory
description with M* as the UV-cutoff.
MP vs. M*
If Einstein gravity theory holds, fundamental gravity coupling does not necessarily
coincide with the Newton constant!!
S grav =
1
4
16G *
[ G * ] = mass
 d x d y | g (4 + n) | R (4 + n) ;
n
-(2+n)
[ G N ] = mass
vs.
-2
The simplest scenario would be considering a flat topology for the extra space, where
the bulk is a factorized manifold of the form M4×T n
m
ds = g mn dx dx
2
n
- d ab dy dy
GN =
a
b
S grav =
G*
Vn
M
2
P
Vn
 d x | g 4 | R4 ;
4
16G *
= M
2+n
*
Vn
ADD, 1998
M* ≈ few TeV ?!!
Notice: Vn encodes the actual geometry of the internal manifold.
M
2
P
= M*
2+ p+q
p
R r
q
Large and short XD
A simple toy model one can consider is an effective field theory on a the torus topology.
R >> r ≥ M*−1
R
3-brane
r
4 brane
  ( y + 2p r ) =  ( y )
M
2
P
= M*
2+ p+q
p
R r
q
Effective Field Theory Prescriptions
To begin with,
• We identify M* as the String scale or Quantum Gravity scale, so bellow such a scale
we can use an effective field theory approach
• We’ll take the brane just as an effective (p+1)D flat surface, inspired on the low
energy limit of a p-brane.
• As for the compact manifold, we will also assume it flat, with some given
coordinates y, such that the brane is localized at some (fixed) point y0
Thus, we need to describe a theory containing fields living on the brane (as SM
fields) and in the bulk (as gravity and perhaps SM singlets), as well as the
interactions among them.
Effective Field Theory Prescriptions
• Bulk fields are described by the higher dimensional action
S bulk   =  d x d y L (  ( x , y ) )
4
n
as S is dimensionless:
[ ] = d +
n
2
• 3-Brane fields, as usual, are described by a four dimensional action, which is
easily promoted into a 4 + n dimensional expression
S brane   =  d x L (  ( x ) ) =  d x d y L (  ( x ) ) d ( y - y 0 )
4
4
n
n
• Brane-Bulk field interactions are localized on space

 d x d y  ( x , y ) ( x ) d ( y - y 0 ) =  d x  ( x , y 0 ) ( x )
4
n
2
n
4
2
Dimensional reduction: 5D toy model
Consider a bulk scalar field φ(x; y).
xm
φ(x; y) = φ(x; y + 2R)
- πR
Thus, φ(x; y) can be Fourier expanded
 ( x, y ) =

1
2p R
 0 ( x) +
0
πR

 ny 
 ny  

(
x
)
cos
+

(
x
)
sin
ˆ




n
 n
R
R
pR 




1

n =1
zero mode
KK modes
S   =
After inserting in the bulk action:
1
2
(
 d x dy    A  - m 
4
A
2
2
)
for A=m,5
we get the effective action:

S =
n =0
1
2
d x
4
(
m
n ) - m 
2
2
n
+ 

2
n
n =1
where the KK mass:
m
2
n
=m +
2
n
2
R
2
1
2
d x
4
(
m
ˆ n ) - m n ˆ n
2
2
2

Expected from PAPA = pm pm + p52 = m2
Dimensional reduction: 5D toy model
 ( x, y ) =

1
2p R
 0 ( x) +
n =1
On the Torus Tn the spectrum would be similar,
but degeneracy on KK levels increases.
•
••
••
•
n=3
n=2
n=1
n=0

b
m
2
n
= m +
2
n
2
R
2

 ny 
 ny  

(
x
)
cos
+

(
x
)
sin
ˆ




n
 n
R
R
pR 




1
1
R2
• E < 1/R: Physics looks 4D
• 1/R < E < M*: up to N~(ER)n KK modes involved.
Evidence of extra dimensions.

Experimental signatures

direct KK production

virtual KK exchange
• E ≥ M*: Effective Field Theory breaks down.
Quantum Gravity regime.
Dimensional reduction: the S1/Z2 orbifold
Take the circle and identify opposite points on it.
xm
- πR
0
πR
Z2
xm
0
Z2 : y → - y
Thus:
- The physical space becomes the interval [0; πR]
-There are two fixed points: y = 0; πR
A scalar field living on this space should now satisfy
the conditions
πR
-periodicity:
φ(x; y) = φ(x; y + 2πR)
- parity:
Z2 φ(x; y) = ± φ(x; y)
U(1)/Z2
n=3
n=2
n=1
n=0
cos(ny/R)
••
•
sin(ny/R)
•
••
even modes
odd modes
Brane to Bulk Couplings
h
 d x dy
4
To get a feeling for the phenomenology, consider:
 ( x , y ) ( x ) ( x ) d ( y )
M*
Using the KK expansion, we get:
h

d
x


 0 +


4
2p R M
*
One gets a suppressed effective coupling:
h eff =
h
Vn M
= h
n
*
M
*
M
P
2
Decay rates:

2 n 
n =1

- Only half of the modes.
- KK modes get an extra √2
- p┴ is not conserved !!
2
Production of a bulk mode
out of brane collisions:

 1
2  M
 h  * 
MP  s

2  M
  h  *  m kk
MP 
φn
KK exchange by brane fields
•Consider the effectuve interaction

2
M = h eff 
1
q -m
2
•By summing up (5D):
2
+ 2
n =1
M = h eff
 

 D (q 2 )
2
2
q - m n 
1
pR
2
q -m
2
2
(
cot p R
•At low energies, q2 << m2 << 1/R2 , we get:
q -m
2
2
M 
h eff
m
2
2
) D ( q
2
)
2

p
2
2
  1 +
m R
3


  D (q 2 )


•At high energies, q2 >> 1/R2, on the other hand:
2
M  2N
h eff
q
2
D (q )
2
D (q ) = h 
2
2
q
2
since N = MR = MP2 / M*2
Bulk to Bulk Coupling Suppression
m
Take for instance the coupling in 5D
on [ -πR, πR ]
 ( x, y )
3
M*
Integrating over the extra dimension we shall get

k , ,m
m
k  m
2 p RM
pR
*
Orthogonality implies
2
2 p RM
*
MP 
=
 
M
* 

 cos
pR

  dy  sin
-pR


( ) cos ( ) cos ( )
( )sin ( ) cos ( )
k y
 y
m y
R
R
R
k y
 y
m y
R
R
R

p
 -
p
2
2
d m , k +  + p d m ,| k -  |
d m , k +  + p d m ,| k -  |

 M* 
 m
  k    |k   |
MP 
k
Fifth momentum is “conserved” in a way that does not
constrain the actual direction of the transverse component
( the sign of p5 is “irrelevant” )
Actually, former p5 conservation is now manifested only as a parity: (-1 )KK
We’ll come back to the orbifold latter…
The Graviton
Giudice, Ratazzi & Wells, NPB 544,3 (1999)
Han, Lykken & Zhang, PRD 59, 105006 (1999)
Take the action for a particle on the brane
S =
and consider the perturbed metric
g MN  h MN + 2 M 1+ n / 2 h MN
1
h
*
MN
 d x | g ( y = 0) | L
4
 h mn + h mn b


An j

Am i 

2 b i j 
hMN is a symmetric tensor, + general coordinates invariance of GR, implies: D(D - 3)/2
independent deg. of freedom
we get, at first order in h the Matter to graviton effective coupling
S int = -
where
1
M
P
 d
4

x G
( n ) mn
n
g T mn = d S
d g mn
-
1
3
2
3(n+2)
b
h
(n)
mn
T
mn
Possible physical processes are:
i) Graviton exchange
ii) Graviton emission
Gravity at short distances
At the classical limit Graviton exchange should provide the law for Gravitational interactions
Existence of extra dimensions could be probed by short distance gravity experiments
U ( r  R )  - G N
m1 m2
r
=U
N
(r )
At large distances, gravity would appear
as effectively four dimensional:
U ( r  R )  - G *
m1 m2
r
n+1
At short distances, however, it would
reveal its higher dimensional nature
Both regimes do match :
G N = G* / Vn
At intermediate scales, just above threshold:
U (r ) = -G N
m1 m2 
-r
1+a e R 


r
But,… How large could R be?...
Simplest Bounds on M* and R
Testing Newton’s law at short distances is not that easy …
r1
F~ GN r1 r2 a4
r2
For: r ~ 20 gr/cm3:
a
r≥a
F ~ 10–5 N × ( a/10 cm )4
This was indeed the strength measured by Cavendish in 1798 !!
C.D. Hoyle et al., hep-ph/0405262
Going to smaller distances must face:
– Surface electrostatic potentials: ~ r −2
– Magnetic forces ~ r −4
– Casimir forces, important for r ~ m m
Sensible to 10–16 N·m
Simplest Bounds on M* and R
Experiments probing short distance gravity have tested Newton’s law down to 160 mm.
No deviations had been found.
Eöt-Was experiment (Washington)
C.D. Hoyle et al.,hep-ph/0405262
Testing for:
U (r ) = -G N
m1 m2 
-r
1+a e R 


r
where, for extra dimensions (r ≥ R )
α= 8n/3
Simplest Bounds on M* and R
Consider
M
2
P
n+2
=M*
R
n
Thus, R < 160 mm or equivalently
1
R
 10 - 3 eV
M* ≥ 1 TeV
On the other hand, from collider physics we know that
For n=1:
If we take R < 160 mm, then
M*
M
= 
 R
2
P




1
3
 10
8
GeV
Notice that if we would rather prefer to take M ~ 1 TeV, say to have a “natural”
solution to the hierarchy problem, thus:
R=
M
2
P
M
3
*
 10
11
m
Simplest Bounds on M* and R
Consider
M
2
P
n+2
=M*
R
n
Thus, R < 160 mm or equivalently
1
R
 10 - 3 eV
On the other hand, from collider physics we know that
M* ≥ 1 TeV
For n=2:
Lets take again R ≈ 160 mm. Now
M*
 M P2
=  2
 R




1
4
 1 . 7 TeV
Might such a low fundamental scale be possible?
In general, for arbitrary n, with M* ~ 1 TeV, one gets
R  2   6

1
n
  10 - 16 + 30 n mm


A solution to the
Hierarchy Problem?
Graviton phenomenology and bounds
Some Processes: Main signal would be energy loss by gravitational radiation into the
bulk by any physical process on the world brane
Single Graviton emission:
Copious production of gravitons in colliders
gKK
Giudice, Ratazzi & Wells, NPB 544,3 (1999)
Han, Lykken & Zhang, PRD 59, 105006 (1999)
f f  V g KK
Feynman diagrams for
During a collision of center mass energy √s there are about N = ( s R )
KK graviton modes!!
Each mode has Planck suppressed couplings
Thus:
 
(
sR
M
2
P
)
n
 s 


 M 


n+2
1
s
S int = -
1
M
P
 d
n
4
n
xG
accesible
( n ) mn
T mn
Graviton phenomenology and bounds
Giudice, Rattazzi, Wells (1999); Mirabelli, Perelstein, Peskin (1999); Han, Likken, Zhang (1999);
Cheung, Keung (1999); Balázs et al., (1999); Hewet (1999)…
LHC:



Single JET:
Drell-Yang:
p+p− → JET gKK
p+p− → ℓ+ℓ− gKK X
Background: ν ν
+
ILC: e e   g KK ;

Z g KK ;
Background: + ν ν
Feynman diagrams for f f  V g KK
H g KK ;
Graviton phenomenology and bounds
Giudice, Rattazzi, Wells (1999)
up to 7 TeV
Missing energy due to graviton emission at LHC, as a function of M*, in a
mono-jet production
Graviton phenomenology and bounds
unpolarized beams
background limit
Total cross section for γ + gKK production at e+ e- linear collider at 1 TeV center mass
energy.
Giudice, Rattazzi, Wells (1999)
Graviton phenomenology and bounds
Abazov, et al., DØ Collab. 2003
95% C.L. exclusion contours
on M* and number of extra
dimensions (n) for monojet
production at DØ (solid lines).
Dashed curves correspond to
limits from LEP, and the
dotted curve is the limit from
CDF, both for γ + gKK
production
Graviton phenomenology and bounds
LHC:

Single JET:
p+p− → JET gKK

Drell-Yang:
p+p− → ℓ+ℓ− gKK X
ILC:
+
e e
-
  g KK ;
Z g KK ;
H g KK ;
Explicit computations of graviton emission leads to some bounds:
Giudice & Strumia, 2003
• Based on Vn=Rn
• Relaxed if one
takes different radii
95% CL limits on M* (in TeV) for n extra dimensions from graviton
emission processes in differente experiments
Graviton phenomenology and bounds
Graviton exchange
gKK
Giudice, Ratazzi & Wells, NPB 544,3 (1999)
Han, Lykken & Zhang, PRD 59, 105006 (1999)
e+ e− → f + f −
e+ e− → γ γ ; W + W − ; ZZ
Bhabha
GG, q q → γ γ ;
Tevatron/HERA:
0.94 TeV
LEP:
0.7 − 1 TeV
LEP:
1.4 TeV
CDF:
0.9 TeV
Giudice & Strumia, 2003
Graviton phenomenology and bounds
J.L. Hewett, 1999
SM Back.
Bin integrated lepton pair invariant mass
distribution for Drell-Yang production for
M*=2.5 and 4.0 TeV at LHC
95 % C.L. search reach for M* as a function of the
integrated luminosity at LHC
Graviton phenomenology and bounds
SM Background
Bin integrated angular distribution (z=cos θ) for
e+ e- → m+ m- and M*=1.5 TeV
95 % C.L. search reach for M* as a function of
the integrated luminosity at e+ e- colliders
J.L. Hewett, 1999
Cosmological bounds
• BBN is very sensible to the expansion rate
n g  n
• At a temperature T there are N= (TR)n
gKK kinematically accessible:
T ~ MeV; n=2; R~0.1 mm  N~1018
•Production rate:
 total 
(TR )
M
n
2
P
=
M*

ng
n

T
n+1
M
M
n+2
T
P
n
n+2
1
*

n+ 2
16 - 3 n
M*
 M *  10 n + 2 GeV
Tr 
MP
M*  10 TeV; n=2  Tr < 100 MeV
n +1
Astrophysical bounds
Hannestad & Raffelt PRD 64 (2001); PRL 88 (2002)
15 - 4 . 5 n
SN 1987a :
M *  10
+
g KK    ; e e
EGRET :
→ 30 TeV ( n=2 )
n+2
-

 g  10
11
 30 MeV

yrs  

m
g 

GRO: M* > 500 TeV
Neutron star heating: M* > 1700 TeV
These bounds:
 Assume no short extra dimensions, and
apply only for R < MeV−1
M
2
P
= M*
2+n
R
n
(M * r ) p
 Assume no graviton decay into ligther KK
modes
gkk → gKK + gKK
Mohapatra, Nussinov & Pérez-Lorenzana, PRD 68 (2003)
Microscopic Black Holes
S.B. Giddings, S. Thomas, PRL 65 (2002) 056010
S. Dimopoulos, G. Landsberg, PRL 87 (2001) 161602
Microscopic Black Hole production
If M* ~ TeV, we may be exploring
effects of Quantum Gravity in LHC/ILC
In (4+n)D Schwarzschild radious:
1
 M BH 
rs ~ 

M
*


1+ n
1
M
*
If impact parameter in a collision is smaller than rS, a BH will form MBH = √s
Cross section:  ~ p rS2 ~ TeV
-2
~ 400 pb
About 107 BH’s per year LHC !!! (?).
Rapid evaporation:
( 3 n +1)
1  M BH 
 


M*  M* 
( n +1)
 10
- 25
seg
KK virtual exchange bounds
A. Mücka, A. Pilaftsis, & R. Rückl,hep-ph/0312186

2
=  ( X ) -  min = 1, 4 , 9
2
2
X =
p
2
3
2
mW r
2