Thermal Unparticles: A New
Form of Energy Density in the
Universe
Nankai University Hu xuepeng
Shao-Long Chen (Taiwan, Natl. Taiwan U.) , XiaoGang He (Taiwan, Natl. Taiwan U. & Nankai U.) , XuePeng Hu, Yi Liao (Nankai U.) . arXiv:0710.5129
1. Original idea due to Georgi
• Unparticle physics
Phys.Rev.Lett.98:221601,2007 （hepph/0703260）
• Another Odd Thing About Unparticle
Physics
• Phys.Lett.B650:275-278,2007
（0704.2457[hep-ph]）
• The very high energy theory contains the fields
of the standard model and the fields(Banks-Zaks
fields) of a theory with a nontrivial IR fixed point.
 The two set interact via heavy particles of mass M U，
1
inducing effective interactions below M U： k OSM OBZ
MU
 Dimensional transmutation occurs at  U in SI sector
 Effective int. below  U： U
4  d SM  dU
OSM OU
• The most important properties of unparticle
• 1 Scale invar. -->the state density of unparticle
d4 p
0
2
2 dU  2
 ( p ) ( p )( p )
4
(2 )
（1）
• 2 the propagator of unparticle is
AdU
i
2 dU
2
2sin( d ) ( p  iò)
dU the scale dimension of the unparticle
operator,is generally non-integral
16 5/2 (dU  1/ 2)
Ad 
(2 ) 2 d (dU  1)(2dU )
U
U
（2）
There has been a burst of activities since the
seminal work of Georgi on various aspects of
unparticle physics:
•
•
•
•
•
FCNC Effects – Precision B physics
collider effects – LHC era
precision QED tests – Any new idea must pass!
unitarity issue of gauge boson scattering
theoretical issues: interpretation of unparticles in
terms of particles
• cosmology and astrophysics
• and so on
2. The thermodynamics of unparticle
• The thermodynamics of a gas of bosonic
particles with mass μ is determined by the
partition function:
4
d
p
2
0
0
ln Z (  )   g sV 
2

2
p

(
p
)
4
(2 )
 ( p   ) ln(1  e
2
2
 p0 
)
（4）
• where  T 1 , gs accounts for degrees
offreedom like spin.
• we can interpret ( 1 ) in terms of a continuous
collection of particles with the help of a spectral
2
2
2 dU  2
functionñ (  )   (  )(  )
:
4
d p
2
2
2 ( p ) ( p   )
ñ
(

)
d

4
(2 )
0
2
2
(4)
 2serves as a new quantum number to be summed
over with the weight ñ (  2 ). Since unparticles exist
only below the scale U , the spectrum must terminate
there,so we can find the normalized spectrum,
ñ (  )  (dU  1)U
2
 (  )(  )
2(1 dU )
2
2 dU  2
• partition function for unparticles is
ln Z  
U 2
0
d  2ñ (  2 ) ln Z (  2 )
g sV (dU  1)
 2 3
4  ( U ) 2( dU 1)

( U )2
0
dy y
dU  2


y
dx x  y ln(1  e  x )
For U  1, the above integrals factorize to good
precision due to the exponential:
g sV
ln Z  3
 ( U ) 2( dU 1)
(6)
with C (dU )  B (3 / 2, dU )(2dU  2) (2dU  2), It is now
straightforward to work out the quantities:
1 
4 T 
pU 
ln Z  g sT 

 V
 U 
2( dU 1)
C (dU )
,
2
4
(7)
2( dU 1)
C (dU )
1 
4 T 
U  
ln Z  (2dU  1) g sT 
(8)

2
V 
4
 U 
The equation of state(EoS) parameter for unparticles:
U 
1
2dU  1
(9)
It is clear that the EoS parameter for unparticles is very different from that for
photons or CDM.The ensemble of unparticles thus provides a new form of
energy density in our universe, which will have important repercussions for
cosmology.
3. unparticle in expanding universe
The unparticle energy density at present is determined by
its initial value at the decoupling temperature TD , and its
evolution thereafter which is closely related to the EoS parameter.
It is given by
3(1 )
 RD 
 ( R)   ( RD )  
( 10 )
 R 
where RD is the scale factor of the expanding universe at decoupling.
Photon expansion follows RD / R  T / TD .we have
 T 
U (T )  U (TD )  
 TD 
 T 
 (T )   (TD )  
 TD 
3(1U )
( 11 )
4
( 12 )
For dU > 1, the unparticle energy density decreases more
slowly than the photon's as the universe cools down.
If unparticle is always in thermal equilibrium with photon,
its energy density drops faster than photon when temperature
goes down. However, after unparticle freezes out of
equilibrium, the situation is different. A dramatic consequence
of this is that even if the unparticle density is small compared
with photon density at a high temperature TD , it may become
larger or even comparable to the critical density at a lower
temperature.
3.1 minimal decoupling temperature
possible interaction between unparticles and SM particle:

UdU
F  F U
the cross section for   U is :
dU
1 2 s  1
 ( s)    2 
AdU
4  U  s
and the interaction rate is ：
 (3) Ad 2  2T 
 n  ( s )c 
 T

2
8
 U 
2 dU
U
when   H the unparticles will decouple from photons，
so we obtains the decoupling temperature：
1  1.66 g
TD  
2  m pl
1/2
*
1/(2 dU 1)
4 

2
 AdU  (3) 

2 dU
U
2
3.2 deof nparticle vs.BBN