Transcript SPACE DIMENSIONS AND UNIFICATION

Extra dimensioni e la
fisica elettrodebole
G.F. Giudice
Napoli, 9-10 maggio 2007
1
SPACE DIMENSIONS AND UNIFICATION
Minkowski recognized special

B
relativistic invariance of

E



E



t
Maxwell’s eqs  connection 

E
between unification of forces 

B

0

B

J


t
and number of dimensions
Electric & magnetic forces unified in 4D space time
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space- time
t, x  x   (t, x )

EM potentialsE   
A
, B   A  A   (, A)
t
EM fields E ,B  F
 0

Ex

  A   A 
E y

E z
current
E x
0
Bz
By
E y
Bz
0
Bx
E z 

By 
Bx 

0 
, J  J   ( , J)
Maxwell's eqs
   F   J 
2
Next step:
UNIFICATION OF EM & GRAVITY
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 New dimensions?
1912: Gunnar Nordström proposes gravity
theory with scalar field coupled to T
1914: he introduces a 5-dim A to describe both
EM & gravity
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1919: mathematician Theodor Kaluza writes a 5dim theory for EM & gravity. Sends it to Einstein
who suggests publication 2 years later
1926: Oskar Klein rediscovers the theory, gives
a geometrical interpretation and finds charge
quantization
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In the ‘80s the theory, known as Kaluza-Klein
3
becomes popular with supergravity and strings
GRAVITY
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In General Relativity, metric g (4X4 symmetric tensor)
dynamical variable describing space geometry (graviton)
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


ds  g dx dx
2
Dynamics described by Einstein action
1
SG 
16 GN
4
d
 x g R(g)
• GN Newton’s constant
•R
curvature (function of the metric)
4
SˆG 
Consider GR in 5-dim
Choose
1
16 Gˆ N
5
d
 x gˆ R(gˆ )
g   2 A A
gˆ MN ( xˆ )  
  A


Dynamical fields

gˆ MN
Assume space is M4S1

  A 
( xˆ )
 
 g , A , 
x5
R
(t,x)
• First considered as a mathematical trick
• It may have physical meaning
5
Extra dim is periodic or “compactified”
x5  2  R  x5
All fields can be expanded in Fourier modes
 n x 5 
 (n ) (x)
 ( xˆ )  
expi

 R 
2 R
n


(n )

(x) Kaluza-Klein modes
5-dim field  set of 4-dim fields:
(n )

Each  has a fixed momentum p5=n/R along 5th dim

extra
dimensions
D-dim
particle


E2 = p 2 + p2extra + m2
4-d space
KK mass
mass
From KK mass spectrum we can measure
the geometry of extra dimensions
6
Suppose typical energy << 1/R 
only zero-modes can be excited
R
r << R
2-d plane
r >> R
Expand SG keeping only
zero-modes and setting =1
1-d line

1
S
(g)

G
16 GN
ˆS (gˆ )  S (g(0) )  S (A(0) ) 

G
MN
G
EM
 S (A)   1
 EM
4
SG
d
4
x g R(g)
4

d
x
F
F


dx5 2 R

1



ˆ
GN
GN
Gˆ N
To obtain correct normalization:

SEM    16 GN
Gravity & EM unified in higher-dim space: MIRACLE?

7
Gauge transformation has a geometrical meaning
g   2 A A
gˆ MN ( xˆ )  
  A

dsˆ 2  gˆ MN ( xˆ ) dxˆ M dxˆ N
  A 
( xˆ )
 
Keep only zero-modes:
dsˆ  g
2

Invariant under local

(0)



dx dx  
x5
A(0)
(0)
dx
5
 A
(0)

dx

 2
 x5  
(where g and 
 A(0)    do not transform)
• Gauge transformation is balanced by a shift in 5th dimension
uniquely determined by gauge invariance
• EM Lagrangian
8
CHARGE QUANTIZATION
Matter EM couplings fixed by 5-dim GR
S
Consider scalar field 
 d xˆ
5
gˆ gˆ MNM  N
2


n

n 2  (n )2 
4
(0)

(0)
(n )

S   dx5   d x g   i
A 
 2

R
R  

n


2
R
Expand in 4-D
KK modes:
Each KKmode n has: mass n/R
n/R
charge

• charge quantization
• determination of fine-structure constant
2
4GN


4 R 2
R2
 R
4GN

 4 10 m  5 10 GeV
31
17
1
• new dynamics open up at Planckian distances
9

Not a theory of the real world
=1 not consistent ( dynamical field leads to
inconsistencies: e.g. F(0)F(0)=0 from eqs of motion)
• Charged states have masses of order MPl
• Gauge group must be non-abelian (more dimensions?)
Nevertheless
• Interesting attempt to unify gravity and gauge interactions
• Geometrical meaning of gauge interactions
• Useful in the context of modern superstring theory
• Relevant for the hierarchy problem?
10
Usual approach: fundamental theory at MPl, while W is
a derived quantity
Alternative: W is fundamental scale, while MPl is a
derived effect
New approach requires
• extra spatial dimensions
• confinement of matter on subspaces
Natural setting in string theory  Localization of gauge theories
on defects (D-branes: end points
of open strings)
We are confined in a 4-dim world,
which is embedded in a higher-dim
space where gravity can propagate
11
COMPUTE NEWTON CONSTANT
Einstein action in D dimensions
1
S 
16 Gˆ N
D
E
D
d
 x gˆ R(gˆ )
Assume space R4SD-4: g doesn’t depend on extra coordinates

Effective action for g
Gˆ N 
1
M DD2
VD4  R D4

VD4
4
SE 
d
x g R(g)

ˆ
16 GN
1 VD4


GN
Gˆ N
M Pl  M D RM D 
D4
2
12
Suppose fundamental mass scale
M Pl  M D RM D 
D4
2
MD ~ TeV
very large if R is large (in units of MD-1)
Arkani-Hamed, Dimopoulos, Dvali
5 104 eV  0.4 mm
D 4  2
20 keV  105  m
1
7
MeV

  30 fm
D 4  4
1
Radius of
compactified space
R
1
D 4  6
• Smallness of GN/GF related to largeness of RMD

• Gravity is weak because it is diluted in a large space
(small overlap with branes)
• Need dynamical explanation for RMD>>1
13
Gravitational interactions modified at small distances
m1m2
FN (r)  GN 2
at r  R
r
At r < R, space is (3+)-dimensional
(=D-4)
(4  ) m1m2
ˆ
FN (r) 
GN

2
r
mm
 GN R 12 2
r
V(r)  GN

m1m2
1  expr 

r
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From SN emission and
neutron-star heating:
MD>750 (35) TeV for =2(3)

14
Testing extra dimensions at high-energy colliders
graviton
gluon
 pp  G
(n )
Probability of producing
a KK graviton
s
jet GN 1028 fb

E2
 2
M Pl
1 event run LHC for1016 tU
than E (use m=n/R)
Number of KK modes with mass less
 n D4  ER
D4
Inclusive cross section
E D4 M Pl2

M DD2
D4

E
 pp  G(n ) jet   sM D2
D
n
It does not depend on VD (i.e. on the Planck mass)

Missing energy and jet with characteristic spectrum

15
16
Contact interactions from graviton exchange
• Sensitive to UV physics
• d-wave contribution to scattering processes
4
• predictions for related processes
T
4T
• Limits from Bhabha/di- at LEP and Drell
1 
1
T  T T  
T T Yan/ di- at Tevatron: T > 1.2 - 1.4 TeV

2 
D2
L
• Loop effect, but dim-6 vs. dim-8
4
L 2 

2


1
  
f   5 f 



2 f  q,l

• only dim-6 generated by pure gravity
•  > 15 - 17 TeV from LEP
17
G-emission is based on linearized gravity, valid at s << MD2
TRANSPLANCKIAN REGIME
Planck length
Schwarzschild
radius
 GD  
3 
c


P  
RS 
1
 2
quantum-gravity scale
1  8
   3 




    2  2 
 GD s 


 c3 


1
 1
 0 :
RS  P
transplanckian limit s  M D :
RS  P
classical limit


1
 1

classical
gravity
same
regime
The transplanckian regime is described by classical physics
(general relativity)  independent test, crucial to verify
gravitational nature of new physics
18
Gravitational scattering
Non-perturbative, but calculable for b>>RS
(weak gravitational field)
b > RS
GD mM
D-dim gravitational potential: V(r)   1
r
D 4 
Quantum-mechanical scattering phase
of wave with angular momentum mvb
1
m v


bc 
GD mM 
b    bc  

b
 b 
 v


b
bc
GD s





L mvb  1 rel.
b 1
E 
4GD s
b
19
Gravitational scattering in extra dimensions:
two-jet signal at the LHC
Diffractive pattern
characterized by
1
GD s 
bc  




20
b < RS
At b<RS, no longer calculable
Strong indications for black-hole formation
BH with angular momentum, gauge quantum numbers, hairs
(multiple moments of the asymmetric distribution of gauge charges and energy-momentum)
Gravitational and gauge radiation during collapse
 spinning Kerr BH
~ RS2
10 pb (for MBH=6 TeV and MD=1.5 TeV)
Hawking radiation until Planck phase is reached
TH ~ RS-1 ~ MD (MD / MBH)1/1)
Evaporation with  ~ MBH(+3)/(+1) / MD2(+2)/(+1)
(10-26 s for MD=1 TeV)
Characteristic events with large multiplicity (<N> ~ MBH / <E>
~ (MBH / MD2)/(+1)) and typical energy <E> ~ TH
Transplanckian condition MBH >> MD ?
21
WARPED GRAVITY
A classical mechanism to make quanta softer
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For time-indep. metrics with g0=0  E |g00|1/2 conserved
.
(proper time d2 = g00 dt2)
Schwarzschild met ric g00  1
2GN M
r

E obs  E em

E em
g00 1  
GN M
rem
On non-trivial metrics, we see far-away objects as red-shifted
22
Consider observer & emitter as 3-d spaces (branes)
embedded in non-trivial 5-D space-time geometries
Randall Sundrum
5th dim S1 / Z2: identify y  y+2R,
y
R
0
y  -y
y
R
0
-y
0
R
Two 3-branes on boundaries &
2
2K|y|
 dx dx  dy2
appropriate vacuum-energy terms  ds  e
23
GRAVITATIONAL RED-SHIFT
ds2  e2K|y| dx dx  dy2
Masses on two branes related by

y=0
y=R
g00=1 g00=e-2RK
mR
 e RK
m0

Same result can be obtained
by integrating SE over y
R  10 K
1

mR
MZ

m0
MGUT
24
PHYSICAL INTERPRETATION
• Gravitational field configuration is non-trivial
• Gravity concentrated at y=0, while our world confined at y=R
• Small overlap  weakness of gravity
WARPED GRAVITY AT COLLIDERS
• KK masses mn = Kxne-RK [xn roots of J1(x)] not equally spaced
• Characteristic mass Ke-RK ~ TeV
• KK couplings
(0)
(n ) 

G
G
L  T  
   
M Pl n1   
   e RK M Pl  T eV
• KK gravitons have large mass gap and are “strongly” coupled
• Clean signal
at the LHC from G  l+l
25
Spin 2
Spin 1
26
A SURPRISING TWIST
AdS/CFT correspondence relates 5-d gravity with
negative cosmological constant to strongly-coupled 4-d
conformal field theory
Warped gravity with
SM fermions and
gauge bosons in bulk
and Higgs on brane
Technicolor-like theory
with slowly-running
couplings in 4 dim
Theoretical developments in extra dimensions have
much contributed to model building of 4-dim theories
of electroweak breaking: susy anomaly mediation,
susy gaugino mediation, Little Higgs, Higgs-gauge
unification, composite Higgs, Higgsless, …
27
What screens the Higgs mass?
boson
fermion
vector
   a
A  A   a
no m 2 2
  e ia 
no m
Spont. broken global
symm.
Chiral
symmetry
Gauge
symmetry
LITTLE HIGGS

SUPERSYMMETRY

5
no m 2 A A 
Symmetry
HIGGS-GAUGE UNIF.
mH
TECHNICOLOR
HIGGSLESS
EXTRA DIMENSIONS
Dynamical EW
breaking
Delayed
unitarity violat.
Fundamental
scale at TeV
• Very fertile field of research
• Different proposals not mutually excluded
Dynamics
28

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M Z2
Necessary t uning 2

M Z2
28

10
2
M GUT
n
Cancellation of
Existence of
electron self-energy
+-0 mass difference
KL-KS mass difference
gauge anomaly
positron

charm
top
cosmological constant
CAVEAT
EMPTOR
a dn a ™emiT kciu Q
ros ser pm oc ed ) de sse rpmoc nU ( FF IT
.e rut cip siht ee s ot de dee n e ra
It is a problem of naturalness, not of consistency!
10-3 eV??
29
HIGGS AS PSEUDOGOLDSTONE BOSON

 f
2
e
i / f
 f
 e :
ia
  

    a
Non- linearly realized symmetryh  h  a forbids m 2 h 2
Gauge, Yukawa and self-interaction are non-derivative couplings
Violate global symmetry and introduce quadratic divergences

➤
Top sector
●
●
➤
No fine-tuning
If the scale of New Physics is so low,
why do LEP data work so well?
30
--
H  a H W a B
LEP1
H  D H
2
iH  D H L   L
LEP2
MFV

e  e 

e    5eb    5b
+
10 9.7
9.2 7.3
1
L 2 O

6.1 4.5
4.3 3.2
2
1
qL u u  q
6.4 5.0

2
H  dR d u u  qL F  9.3 12.4
new
physics
Bounds on  [TeV]
5.6 4.6

Little Higgs
strong
Composite Higgs
dynamics
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energy
1 TeV
10 TeV
A less ambitious programme: solving the little hierarchy
31
LITTLE HIGGS
Explain only little hierarchy
One loop m 
2
H
GF

2
m 
2
SM
2
SM
  SM 

GF
 T eV
At SM new physics cancels one-loop power divergences
T wo loops mH2 
GF2

4
4
mSM
2   
2
GF mSM
 10 T eV   LH
“Collective breaking”: many (approximate) global symmetries
preserve massless Goldstone boson
ℒ1
ℒ2
H
ℒ1 ℒ2 2
m  2

2
4 4
2
H
32
It can be achieved with gauge-group replication
•Goldstone bosons in G / H
• G  G1  G2 gauged subgroups, each preserving
a non-linear global symmetry
• SM  G1  G2 which breaks all symmetries
Field replication Ex. SU2 gauge with 1,2
doublets such that V(11,22) and 1,2
spontaneously break SU2
Turning off gauge coupling to 1 
Local SU2(2) × global SU2(1) both spont.
4
g
broken 2
2
mH 
4 
4

twoloops
33
Realistic models are rather elaborate
Effectively, new particles at the scale f cancel
(same-spin) SM one-loop divergences with
couplings related by symmetry
Typical spectrum:
Vectorlike charge 2/3 quark
Gauge bosons EW
triplet + singlet
Scalars (triplets ?)
34
New states have naturally mass
New states cut-off quadratically divergent contributions to mH
Ex.: littlest Higgs model
Log term:
analogous to effect of stop
loops in supersymmetry
Severe bounds from LEP data
35
TESTING LITTLE HIGGS AT THE LHC
• Discover new states (T, W’, Z’, …)
• Verify cancellation of quadratic divergences
f from heavy gauge-boson masses
mT 2t  2T

f
2T
mT from T pair-production
T : we cannot measure TThh vertex
(only model-dependent tests possible)
36
f and gH
from DY of
new gauge
bosons
Production rate and BR
into leptons in region
favoured by LEP (gH>>gW)
Can be seen up to ZH mass of 3 TeV
MT from T production can be
measured up to 2.5 TeV
37
T  bW   2T  tZ  2T  th  2T
Measure T width?
Cleanest peak from
In order to precisely extract T from measured cross
section, we must control b-quark partonic density
Possible to test cancellation with 10% accuracy
for mT < 2.5 TeV and mZ < 3 TeV
38
NEW INGREDIENTS FROM EXTRA DIMENSIONS
HIGGS AS EXTRA-DIM COMPONENT OF GAUGE FIELD
AM = (A,A5),
gauge
A5 g A5 +∂5 
Higgs
forbids m2A52
Higgs/gauge unification as
graviton/photon unification in KK
Correct Higgs quantum numbers by projecting out
unwanted states with orbifold
The difficulty is to generate Yukawa and quartic
couplings without reintroducing quadratic divergences
39
HIGGSLESS MODELS
Breakdown of unitarity: 4 d    4 mW  T eV
g
24 3 12 2 mW
5d    2 
 10 T eV
2
g5
g
The gauge KK modes delay unitarity violation

New ways of breaking
gauge symmetries:
no zero modes in restricted extra-D spaces
(Scherk-Schwarz mechanism)
40
y
Scherk-Schwarz breaking
R
A field under a 2R translation has to remain the same,
unless there is a symmetry (the field has to be equal up
to a symmetry transformation)
x, y  2R  e iQ x, y 
KK expansion with orbifold boundary condition
x, y   e
2


Q
1
(n )
2
e

(x)

m

n


 2

n
 2  R
n
iQy 
2 R
iny
R
No more zero-modes!
41
At the LHC discover KK resonances of gauge bosons
and test sum rules on couplings and masses required
to improve unitarity
42
DUALITY
SM in warped extra dims  strongly-int’ing 4-d theory
KK excitations  “hadrons” of new strong force
Technicolor strikes back?
TeV brane
Planck brane
5-D gravity
5th dim
4-D gauge theory
Motion in 5th dim
UV brane
IR brane
RG flow
IR
AdS/CFT
RG flow
Planck cutoff
breaking of conformal inv.
Bulk local symmetries global symmetries
UV
5-D warped
gravity

large-N
technicolor
 Composite Higgs
43
Technicolor-like theories in new disguise
Old problems
TC
The presence of a light Higgs helps
• Light Higgs screens IR contributions to S and T
N v2
S
•
(f pseudo-Goldstone decay constant)
6 f 2
44
Can be tuned small for strong dynamics 4f at few TeV
Structure of the theory
quarks, leptons
& gauge bosons
strong
sector
Communicate via gauge (ga)
and (proto)-Yukawa (i)
Strong sector characterized by mmass of resonances
g coupling of resonances
Take I, ga << g < 4
In the limit I, ga =0, strong sector contains
Higgs as Goldstone bosons
Ex. H = SU(3)/SU(2)U(1) or H = SO(5)/SO(4)
-model with f = m/ g
45
ga , i break global symmetry  Higgs mass
New theory addresses hierarchy problem  reduced
sensitivity of mH to short distances (below m-1)
 2
m 
m
4
2
H
Ex.:

• Georgi-Kaplan:
g=4, f = v, no separation of scales
• Holographic Higgs: g= gKK, m= mKK
• Little Higgs: g, mcouplings and masses of new t’, W’, Z’
46
Production of resonances at m allows to test models
at the LHC
Study of Higgs properties allows a model independent
test of the nature of the EW breaking sector
Is the Higgs
fundamental?
composite?
SM (with mH < 180 GeV)
Holographic Higgs
supersymmetry
Gauge-Higgs unification
Little Higgs
47
Construct the Lagrangian of the effective theory below m
U  e i T Goldstones;  heavy f ields

2

 
m4  (0)
g4
  g (1)
 

(2)

 2 L 
U,,

L
U,,

L
U,,

...





 16 2 


 
2 2
g  
m
m
m
16

 

  
 
 
 

a
LSILH
a
From the kinetic term, we obtain the definition of f = m / g
 Each extra H insertion gives operators suppressed by 1 / f
• Each extra derivative
“
“

g2
4  q2
 2 1 2  ...
2
2
mW  q
v  mW

f: symmetry-breaking scale
1 / m
m: new-physics mass threshold
• Operators
that violate Goldstone symmetry are suppressed

48
by corresponding (weak) coupling
Operators testing the strong self coupling of the Higgs
(determined by the structure of the  model)

cH  
c 6   3 c y y f 
1


H
H

H
H

H
H

H
Hf
Hf

h.c.




 2


L
R
2
2 
2f
f
f2
 f

 and yf are SM couplings; ci model-dependent coefficients
Form factors sensitive to the scale m
icW g  i 
ic B g  

i

H

D
H
D
W

H
D
H

B

2
2
2m
2m





1
m 2
2 2
c g2 g 2 
c
g
g S yt


a
a
H
HB
B

H
HG
G


16 2 m2
16 2 m2
Loop-suppressed strong dynamics

 i
ic HW g
ic HB g   


i

D H   D H W  
D H  D H B
2 2 
2 2 
16 f
16 f
g2 1
16 249m2
Effects in Higgs production and decay
v2 1

f2 4

2
1 
v 2  h   
c H  L  1 c H 2 1   h h All Higgs couplings
2 
f  v  
rescaled by
m 
v 2 
c y  L   1 c y 2  h
v 
f 
1
v2
1 c H 2
f
cH v2
 1
2 f2
Modified Higgs couplings to matter

50
Dührssen 2003
SLHC Report 2002
51
LHC can measure cHv2/f2 and cyv2/f2 up to 20-40%
SLHC can improve it to about 10%
A sizeable deviation from SM in the absence of new light states
would be indirect evidence for the composite nature of the Higgs
ILC can test v2/f2 up to the % level
ILC can explore the
Higgs compositeness
scale 4f up to 30 TeV
ECFA/DESY LC Report 2001
52
• Effective-theory approach is half-way between modeldependent and operator analyses
• Dominant effects come from strong self-Higgs interactions
g 1
characterized by

m
f
• From operator analyses, Higgs processes loopsuppressed in SM are often considered most important for
searches

• However, operators h and hgg are suppressed
1/(162m2)
• Since h is charge and color neutral, gauging SU(3)cU(1)Q
does not break the generator under which h shifts
(Covariant derivative acting on h does not contain  or g)
• Not the case for hZ (loop, but not 1/g2 suppressed)
53
Higgs decay rates
54
Genuine signal of Higgs compositeness at
high energies
In spite of light Higgs, longitudinal gauge-boson
scattering amplitude violate unitarity at high energies
WL
WL
h
Modified coupling
LHC with 200 fb-1 sensitive up to cH 0.3
WL
WL
55
Higgs is viewed as pseudoGoldstone boson:
its properties are related to those of the exact
(eaten) Goldstones: O(4) symmetry
Strong gauge-boson scattering
 strong Higgs production
Can bbbb at high invariant mass be separated from background?
h  WW  leptons is more promising
Sum rule (with cuts ||and s<M2):
56
In many realizations, the top quark belongs to
the strongly-coupled sector
At leading order in 1/f2
Modified top-quark couplings to h and Z
At ILC ghtt up to 5% with s=800 GeV and L=1000 fb-1
From gZtt, cR ~ 0.04 with s=500 GeV and L=300 fb-1
FCNC effects
57