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“Understanding confinement”, May 16-21, 2005
Quantum Black Holes,
Strong Fields,
and
Relativistic Heavy Ions
D. Kharzeev
Outline
• Black holes and accelerating observers
• Event horizons and pair creation in
strong fields
• Thermalization and phase transitions
in relativistic nuclear collisions
DK, K. Tuchin, hep-ph/0501234
Black holes radiate
S.Hawking ‘74
Black holes emit
thermal radiation
with temperature
acceleration of gravity
at the surface
Similar things happen in
non-inertial frames
Einstein’s Equivalence Principle:
Gravity
Acceleration in a non-inertial frame
An observer moving with an acceleration a detects
a thermal radiation with temperature
W.Unruh ‘76
In both cases the radiation is
due to the presence of event horizon
Black hole: the interior is hidden from
an outside observer;
Schwarzschild metric
Accelerated frame: part of space-time is hidden
(causally disconnected) from
an accelerating observer;
Rindler metric
From pure to mixed states:
the event horizons
mixed state
pure state
mixed state
pure state
Thermal radiation can be understood
as a consequence of tunneling
through the event horizon
Let us start with relativistic classical mechanics:
velocity of a particle moving
with an acceleration a
classical action:
it has an imaginary part…
well, now we need some
quantum mechanics, too:
The rate of tunneling
under the potential barrier:
This is a Boltzmann factor with
Accelerating detector
• Positive frequency Green’s function (m=0):
• Along an inertial trajectory
• Along a uniformly accelerated trajectory
x  y  0,z  (t 2  a2 )1/ 2
Unruh,76
detector is effectively immersed into a heat bath
Accelerated
at temperature TU=a/2
An example: electric field
The force:
The acceleration:
The rate:
What is this?
Schwinger formula for the rate of pair production;
an exact non-perturbative QED result
factor of 2: contribution from the field
The Schwinger formula
eE
e+
Dirac sea
+
+
+…
The Schwinger formula
• Consider motion of a charged particle in a constant
electric field E. Action is given by
Equations of motion
yield the trajectory
where a=eE/m is
the acceleration
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Classically forbidden
trajectory t  it E
• Action along the classical trajectory:


m
eE
S(t)   arcsh(at)  2 at( 1 a2t 2  2)  arcsh(at)
a
2a
• In Quantum Mechanics S(t) is an analytical function of t
m eE m 2
 • Classically forbidden paths contribute to ImS(t) 
 2 
a
2a
2eE
• Vacuum decays with probability
V m  1 exp e
2 Im S
 e
2 Im S
 m 2 / eE
e
Sauter,31
Weisskopf,36
Schwinger,51
• Note, this expression can not be expanded in powers
of the coupling - non-perturbative QED!

Pair production by a pulse
Consider a time dependent field


E
A  0,0,0, tanh( k0 t)
k0


E

• Constant field limit k0  0
• Short pulse limit
k0  
 spectrum with
a thermal

t
Chromo-electric field:
Wong equations
• Classical motion of a particle in the external nonAbelian field:
 gfx
Ý  gFIÝq
Ý IxÝaA
mxÝ

a
abc

I 0
b c
The constant chromo-electric field is described by
A0a  Ez a3, Ai a  0
 vector I3 precesses about 3-axis with I3=const
Solution:
0
Ý
Ý
Ý
Ý
Ý
Ý
Ý
x  y  0,mz  gEx I3

Effective Lagrangian: Brown, Duff, 75; Batalin, Matinian,Savvidy,77

An accelerated observer
consider an observer with internal degrees of freedom;
for energy levels E1 and E2 the ratio of occupancy factors
Bell, Leinaas:
depolarization in
accelerators?
For the excitations with transverse momentum pT:
but this is all purely academic (?)
Can one study the Hawking radiation
on Earth?
Gravity?
Take g = 9.8 m/s2;
the temperature is only
Quic kTime™ and a
TIFF (Unc ompres sed) dec ompres sor
are needed to see this pic ture.
Electric fields? Lasers?
compilation by A.Ringwald
Condensed matter black holes?
“slow light”?
Unruh ‘81;
e.g. T. Vachaspati,
cond-mat/0404480
Strong interactions?
Consider a dissociation of a high energy hadron of mass m into
a final hadronic state of mass M;
The probability of transition:
m
M
Transition amplitude:
In dual resonance model:
Unitarity:  P(mM)=const,
b=1/2
universal slope
Hagedorn
temperature!
limiting acceleration
Fermi - Landau statistical model
of multi-particle production
Hadron production
at high energies
is driven by
statistical mechanics;
universal temperature
Enrico Fermi
1901-1954
Lev D. Landau
1908-1968
Where on Earth can one achieve the largest
acceleration (deceleration) ?
Relativistic heavy ion collisions! stronger color fields:
Strong color fields
(Color Glass Condensate)
as a necessary condition for
the formation of Quark-Gluon Plasma
The critical acceleration (or the Hagedorn temperature)
can be exceeded only if the density of partonic states
changes accordingly;
this means that the average transverse momentum
of partons should grow
CGC
QGP
Quantum thermal radiation at RHIC
The event horizon
emerges due to the fast
decceleration
of the colliding nuclei
in strong color fields;
Tunneling through
the event horizon
leads to the thermal
spectrum
Rindler and Minkowski spaces
The emerging picture
Big question:
How does the produced
matter thermalize
so fast?
Perturbation theory +
Kinetic equations
Fast thermalization
horizons
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
collision point
Qs
Rindler coordinates:
Gluons tunneling through the event horizons have thermal
distribution. They get on mass-shell in t=2Qs
(period of Euclidean motion)
Deceleration-induced phase transitions?
• Consider Nambu-Jona-Lasinio model in Rindler space


L  i (x) (x) 
2N
2
2
(

(
x)

(x))

(

(
x)i


(x))


5
• Commutation relations:
  (x), (x) 2g (x)

• Rindler space:
  x t ,
2

2
2
1
2
  ln
tx
tx
ds2   2d2  d 2  dx


2
e.g., Ohsaku ’04
Gap equation in an accelerated frame
• Introduce the scalar and pseudo-scalar fields

 (x)   ( x)(x)
N

 (x)   ( x)i 5(x)
N
• Effective action (at large N):


Seff
  2   2 

  d x g
 iln det(i     i 5 )
 2  
4
• Gap equation:

2i
 
a

d 2k 
sinh(  /a)
2
2
d

(K
(

/a))

(K
(

/a))
)


i

/
a
1/
2
i

/
a1/
2
2
2
(2 ) 

where


 2  k 2   2
Rapid deceleration induces phase transitions
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
NambuJona-Lasinio model
(BCS - type)
Similar to phenomena in the vicinity of a large black hole:
Rindler space
Schwarzschild metric
A link between General Relativity and QCD?
solution to some of the RHIC puzzles?
Black holes
RHIC collisions