Making the most and the best of Unparticle
to gain as much advantage and enjoyment as you can from sth
to accept a difficult situation and do as well as you can
A survey of scale invariance and conformal group
Banks-Zaks fields as an example of non-trivial IR fixed point
Unparticle effective theory, propagator and vertex
The difficulty in giving unparticle SM gauge quantum number
Scale invariance breaking
Gauge interaction of unparticle
It has been a long dream that the current human size is
not so special.
It will be business as usual after a scale transformation.
Scale Transformation
M  3 M


x  x
A  2 A
The weight a unit area of bone could sustain has to change, too.
Scale transformation in QFT
x   x 
Fundamental particle masses will break scale symmetry
In high energy, the masses of all particles may be ignored and a
scale invariant theory will emerge.
From studying scale invariant QFT, like massless free field, physicists
found the theory is invariant under Minkowski space inversion I
I:x
x
x2
From translation T T : x  x  a
x  a x2
ITI : x 
1  2 x a  a 2 x 2

we can generate anew symmetry.
Special conformal transformation
Conformal group: Transformations that preserve the form of the
metric up to a factor. g ( x)  2 ( x)  g ( x)


It preserve the angle between two 4 vectors
A  B
A2 B 2
It includes Poincare transformations, scale transformation and …
x  a x2
x 
1  2 x a  a 2 x 2

Special Conformal Transformation
It is widely believed that unitary interacting scale invariant theories are
always invariant under the full conformal group. (only proven in 2D)
S. Coleman et al have shown that under some conditions, a scale
invariant theory is also comformally invariant (including all renormalizable
field theory)
The properties of a scale invariant theory are usually determined by a set
of operators which is eigenfunctions of the scaling operator D.
O( x)  O' ( x)  du  O(x)
Conformal invariance severely restricts the two point function of these
operators.
O(0)O( x) 
1
x 
2 du
They are not eigenfunction of mass operator P2.
They have a continuous spectrum.
D, P   2iP
2
2
0
In QFT, it is more complicated due to the presence of renormalization
scale 
.
The coupling constant depends on μ.
For dimensionless g
g  f (ln


)
we get a dimensionful parameter Λ.
Scale invariance is broken.
unless ……. at some point
dg

d ln 
Dimensional transmutation
 0
Fixed Point
At fixed point, scale invariance is recovered.
Field theories generally exhibit scale invariant UV fixed point (often free)
and scale invariant IR fixed point (often trivial, meaning non-interacting).
What if the IR fixed point is non-trivial?
For an SU(3) gauge theory with NF massless Dirac fermion
3
5


g
g


 ( g )   0
 1
2 2 
 16 2
16  

4
3
 0  11  T ( R) N F
   at
NF  N * 
33
4T
1  102  20  4C2 ( R)T ( R)  N F
   at
N F  N 'F 
102
20  4C2 T
N 'F  N *
If N *  N F  N 'F
 0  0, 1  0
Two loop β function has a non-trivial zero.
This asymptotic free
gauge theory with
massless fermions has a
g*
non-trivial IR fixed point
UV
IR
 ,   0, g  constant
Scale invariance
Banks-Saks (BS) Model
BZ Model
g  g * c  log
  U
Scale invariant theory

U
Dimensional g  
U
transmutation
If it becomes strong interacting near IR fixed point, U   BS
massless fermions
BZ Model
OBZ
  U
Scale invariant theory
d BZ  dU
U

OU
Integrate out
degrees of
freedom which is
usually of order U or  BS
operators which are eigenfunctions of
scale transformation D
O( x)  O' ( x)  d  O(x)
u
OU is of dimension du
They will stay since they have
continuous spectra.
unparticles
Unparticle propagator
Scale invariance almost determines unparticle propagator completely.
0 OU ( x) O (0) 0   e

U
insert 1  
n
iPx
0 OU (0) P
2
d 4P
 (P )
2 4
2
d 4P
n n 
 ( P) P P
2 4
Scale invariance dictates the left scale with dimension 2dU.
2
 
0 OU (0) P  ( P 2 )  AdU P 2
dU  2
This is identified as the phase space factor for n massless final particles.
4
n
d
pj


4
2
0


2    P   p j   p j  p j
 An P 2
3
2 
j 1

 j 1
4
  
 
n2
Unparticle with dimension dU looks like a non-integer number dU of particles.
 
0 OU ( x)O (0) 0   e AdU P

U
iPx
 dM  (M

2

2

 AdU  dM  M 2

0
2

2 dU  2
 
 P )  e AdU P
2
 
dU  2
d 4P
2 4
iPx
2 dU  2
d 4P
2 4
Unparticles have
continuous spectra
of masses.
d 4 P iPx
2
2 
e

(
M

P
)
 2 4

The usual factor
for a normal field
operator
take time order and Fourier transformation


4
iPx

d
x
e
0
T
O
(
x
)
O
U
U (0) 0 


 
 2

dM
M


2 0

AdU
i
AdU
2

dU  2

i

P 2  M 2  i 
1
 P 2  i
2 sin dU  

dU  2
x p 1

dx

0 1  x sin p
well defined for
negative P2
For non-integer dU there is a cut
in the space of P2.
The cut has to chosen at positive timelike P2.
The cut could be seen as a combination of continuous poles.
P2
This is really no surprise since unparticle has continuous mass spectrum.
Interaction with SM particles
through the exchange of a heavy particle of mass MU
1
OSM  OBZ
M Uk
  U
dUBZ  dU
CU
OSM  OU
M Uk
Non-renormalizable vertex
With unparticle vertex and propagator, very
interesting phenomenology can be studied.
K. Cheung, W.Y. Keung and T.C. Yuan PRD 76 055003
C.H. Chen and C.Q Geng, PRD
continuous missing energy in real unparticle emission
Z  qq  U
K. Cheung, W.Y. Keung and T.C. Yuan, PRL
interference with SM through virtue unparticle exchange
Drell-Yan Process
Unparticles are a hidden sector, like heavenly god
U
OSM OU
MU
They are so unlike us, normal, earthly particles.
Seeing them needs to be so rare that it’s called a miracle.
To increase its importance in our earthly life and to teach us his
message, God needs an incarnation, ie. becoming a human form.
Unparticles needs to be given SM gauge quantum numbers.
That miracle still doesn’t happen everyday means that scale
invariance needs to be broken in the low energy and will
manifest itself as energy gets higher.
Scale invariance most likely will be broken anyway.
P. Fox, A. Rajaraman and
Y. Shirman PRD 2007
1
M
d BZ  2
U
4  dU
U

dUBZ  dU
2
CU d BZ  2 H  OU
MU
H  OBZ
2
 U
 
 MU



v
d BZ  2
 2U dU  v 2
The scale invariance supposedly will be broken at U
conformal window U  E  U
For the window to be not too narrow, M  U
Nonrenormalizable couplings will be suppressed.
Unparticles will be unaccessible.
U  v, M  10v
U  v, M  2v
M. Bander, J. Feng, A.
Rajaraman and Y.
Shirman 0706.2677
However, we don’t need non-renormalizable terms to access
unparticle in case they are incarnated, ie. has SM quantum number.
Imagine the following scenario:
U  v
Make sure we didn’t see unparticle until LHC
U  10 2  v
The conformal window is about two degrees of magnitude
M U  103  v
Non-renormalizable interaction could be ignored.
We ask the same question Howard asked:
How does a SM flavored or colored unparticle look like in collider?
Two hurdles to overcome:
How do we introduce scale invariance breaking effects?
How do we flavor or color a unparticle?
How do we introduce scale invariance breaking effects?
Parameterize the breaking with an infrared cutoff.
Parameterize the breaking with an infrared cutoff m.
( p, m, dU )   d 4 x  e ipx 0 TO( x)O  (0) 0
A
 U
2


2
2
2
dM

M

m


dU  2
m2
i
p 2  M 2  i
It reduced to Georgi’s unparticle propagator as
m0
and reduced to particle propagator with mass m as
dU  1
P2
( p, m, dU ) 
AU
i
2 sin dU  p 2  m 2  i


2  dU
m2
How do we flavor or color a unparticle?
Gauge interaction of unparticles
How do we flavor or color a unparticle?
Gauge interaction of unparticles
The unparticle propagator naively will imply a non-local Lagrangian:
2 sin dU 
S
AdU

d4p 
2
2
 2 4  ( p) m  p

2 dU
 ( p)
S   d 4 x d 4 y   ( y ) F ( x  y )  ( x)
For gauge symmetry,
insert a Wilson Line
y


a
a

W ( x, y )  P exp  igT  A dw 
x


Vertex of a Gluon coupled to two unparticles.
W ( x, y)  U ( x)W ( x, y)U  ( y)
Vertex of two gluons and two unparticles
qq  g*  Unpartcile s
U
 2  dU
P
Corresponding scalar particle
pair production cross section
Option II My suggestion is to use the representation of unparticle as
bulk field in extra dimensional model
d
4
xe
iPx
dM 2
i
0 TO( x)O(0) 0  
O ( M 2 ) 2
2
P  M 2  i
The unparticle propagator contains a cut for timelike P.
A cut line can be decomposed into a collection of point
poles with the gap goes to zero!
An unparticle may correspond to a collection of particles!
d
4
xe
iPx
dM 2
i
0 TO( x)O(0) 0  
O ( M 2 ) 2
2
P  M 2  i
 (M 2 )  Ad (M 2 )d
Scale invariance
u 2
u
It suggests a collection of non-interacting particles created by operator
O with continuous mass distribution. Start with a discrete form:
 0
M n    f (n)
d
4
xe
iPx
mass gap
iFn2
0 TO( x)O(0) 0   2
2
n P  M n  i


Fn2   0 O(0)   M n2  M 2  2
2

 
Scaling F  M
2
n
2 du 2
n
Towers of particles appear in extra dimensional models.
Kaluza-Klein Modes of Bulk Field
•
•
Bulk Fields contains Kaluza-Klein
n
i y
(KK) States Ψn with wavefunctions: e R
due to periodicity.
The extra-dimension wave number k  n / R
•
•
KK states look like having 4D
masses n
2
…….
2
E-p relation E  p   n   0
R
2
R

n2 

     5 5 n       2  n  0
R 




Kaluza-Klein Modes of Bulk Field
…….
R  ,   0
Non-compact extra dimensions gives towers of
particles with continuous mass distribution.
Assume only deconstructed unparticle see the
non-compact extra dimension. Even gravity won’t
see it.
Or the gravity is localized!
RSII
Scaling


Fn2   0 O(0)   M n2  M 2  2
2

 
Fn2  M n2
du 2
needs to scale with Mn2
ADD realization
Consider ADD with m extra dimensions and a bulk scalar field ( x, y1 , y2  ym )
Assume that O( x)   ( x,0,0,  0)

is a KK state
0 O(0)   1
Fn2    M n2  M 2   2  N M n2   2
density of states

 ni 
For m extra dimensions M    R 

i 1 
m
2
2
n
Density of states is proportional to the hyper sphere shell:
 
N (M )  M
2
n
2
n
m1
2
 
d M
2 1/ 2
n
 
 M
m
2 2 1
n
 
 d M n2
 
F  M
2
n
du 
m
1
2
W.Y. Keung
du  1.5, 2.0, 2.5......
2 du 2
n
ADD realization of deconstruction
Bulk scalar field in 5 noncompact dimensions
unparticle with dimension 3/2
need a ultraviolet cutoff
Bulk scalar field in 4 non-compact continuous dimensions and one noncompact discrete dimension
ADS-CFT
Consider one extra dimension: z
Need 4D Poincare symmetry:

ds 2  w( z) 2  dx  dx  dz 2
4D unparticle theory is scale invariant

x  x
In 5D, no longer conformal, it needs to corresponds to an isometry
of the metric.
g g
MN
z  z
ds 2 
  dx  dx  dz 2
z
2
R2

MN
R
z
AdS5 Anti-de-Sitter Space
AdS5 Anti-de-Sitter Space: the most symmetric spacetime with negative curvature
ds 2  e2ky dx  dx  dy 2
Conformally Flat frame
e ky
z
k
ds 
2
  dx  dx  dz 2
z
2
dz  e ky dy
R
g~MN  z 2 g MN   MN
2
R
1
k
It’s common to take out
the dimensions from all
the coordinates.
yz
y    z  0
Non-compact
Boundary
ds 2 
  dx  dx  dz 2
z
2
Isometry in AdS5
x   x, z   z
The metric does not change
g MN  g MN
Conformal Symmetry on the boundary z=0
x  x
CFT
R2
AdS-CFT Correspondence
Non-compact AdS5 model
0
unpartcile CFT in 4D
 ( x, z )
What is the dimension of the unparticle that
corresponds to a massive bulk scalar field?
Bulk scalar field in AdS5


S   dx 4 dz g g MN  M  N   m52  2 / 2
Boundary
operator in
unparticle CFT
O( x)  lim z  du ( x, z)
Dimension du is the
solution of Eq.
z 0
m52  d u (d u  4)
Bulk scalar field correlation function in AdS5


S   dx 4 dz g g MN  M  N   m52  2 / 2
dx 2  dz 2 2
ds 
R
2
z
2

dx 4 dz 2
S  N  5 z  M  M   m52 R 2 2
z
2

Two point function
O( x)O(0)  lim z  du z '
 du
z 0
( x, z )(0, z ' )  lim z  du z '
 du
z 0
G( x; z, z ' )
  3 
3 2
5 2 
z

z
q

z
m5 G (q; z , z ' )   ( z  z ' )
 z
z

If we care only the limit z  0
 5  3 
2
z
z

m
5  G ( q; z , z ' )  0
 z
z

G  z
m52   (  4)
du
O( x)O(0)  lim z  du z '
 du
z 0
G( x; z, z ' )
under scale transformation x  x
O( x)O(0)  O(x)O(0)  lim z  du z '
 du
z 0
G(x; z, z ' )
x  1 x, z  1 z
Isometry in AdS5
   z'
 lim 2du 1 z
z 0
 du
 2 d u O( x)O(0)
1
 du
G( x; 1 z, 1 z' )  2du lim Z
 du
Z0
O( x)O(0)  x 2 d u
Z'd
u
G( x; Z, Z' )
 
z
G( z, x , x ' )   
2
2 
 x  x '  z


O( x)O(0)  x 2 d u
 1
z

Consider the compact version
0,  
KK expasion
G (q; z , z ' )  
n
n ( z )  z d
For large n
n  z d
u 1
Mass spectrum M n 
n ( z )n ( z ' )
( x, z )    n ( x) n ( z )
q 2  M n2  i
n
u
  3 
3
2
5 2 
z

z
M

z
m5 n ( z )  0
n
 z
z

sin M n z  C1 
n  C2
zm
Same thing can be done for other Lorentz structure (there is a table
of relations between dimensions and bulk masses in the Maldacena
et al review)
Deconstruction
Non-compact AdS5 model
0
unpartcile CFT in 4D
 ( x, z )
Non-exotic, but with gravity!
1
k
An ultraviolet brane corresponds to unltraviolet cutoff in 4D.
To localize gravity, may need a Planck brane at z 
RSII
Unparticle theory cutoff at M u
Gauge interaction of unparticles
Non-compact AdS5 model
0
unpartcile CFT in 4D
 ( x, z )
introduce bulk color
gauge field and bulk
gauge transformation
put SM on the brane,
transforming under
restricted localized
gauge transformation
Bulk scalar field can be
naturally colored or
flavored!
a bonus
0
 ( x, z )
IR brane introduce a natural
infrared cutoff
An IR brane will naturally give an IR cutoff and hence a
breaking of scale invariance
Other deconstruction
( flat without gravity )
Fractal Theory Space
Fractional Dimension