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Quantum Black Holes, Strong Fields, and Relativistic Heavy Ions
D. Kharzeev
“Understanding confinement”, May 16-21, 2005
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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
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Black holes radiate
Black holes emitthermal radiationwith temperature
S.Hawking ‘74
acceleration of gravityat the surface
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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 detectsa thermal radiation with temperature
W.Unruh ‘76
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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
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From pure to mixed states:the event horizons
mixed state pure state mixed state pure state
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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…
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well, now we need some quantum mechanics, too:
The rate of tunnelingunder the potential barrier:
This is a Boltzmann factor with
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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
Accelerated detector is effectively immersed into a heat bath at temperature TU=a/2
Unruh,76
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An example: electric fieldThe 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
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The Schwinger formula
e+
e-
Dirac seaE
+ + + …
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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 QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
where a=eE/m isthe acceleration Classically forbidden
trajectory
€
t → −itE
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€
S(t) = −m
aarcsh(at) +
eE
2a2at( 1+ a2t 2 − 2) + arcsh(at)( )
• Action along the classical trajectory:
• In Quantum Mechanics S(t) is an analytical function of t
• Classically forbidden paths contribute to
€
ImS(t) =mπ
a−
eEπ
2a2=
πm2
2eE
• Vacuum decays with probability
€
ΓV →m =1− exp −e−2 Im S( ) ≈ e−2 Im S = e−πm 2 / eE
• Note, this expression can not be expanded in powers of the coupling - non-perturbative QED!
Sauter,31Weisskopf,36Schwinger,51
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Pair production by a pulse
€
Aμ = 0,0,0,−E
k0
tanh(k0t) ⎛
⎝ ⎜
⎞
⎠ ⎟
Consider a time dependent field E
t
• Constant field limit
€
k0 → 0
• Short pulse limit
€
k0 → ∞
a thermal spectrum with
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Chromo-electric field:Wong equations
• Classical motion of a particle in the external non-Abelian field:
€
m˙ ̇ x μ = gF qμν ˙ x ν Ia
€
˙ I a − gfabc ˙ x μ AμbI c = 0
The constant chromo-electric field is described by
€
A0a = −Ezδ a3, A i
a = 0
Solution: vector I3 precesses about 3-axis with I3=const
€
˙ ̇ x = ˙ ̇ y = 0,m˙ ̇ z = gE˙ x 0I3
Effective Lagrangian: Brown, Duff, 75; Batalin, Matinian,Savvidy,77
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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 inaccelerators?
For the excitations with transverse momentum pT:
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but this is all purely academic (?)
Take g = 9.8 m/s2; the temperature is only
Can one study the Hawking radiationon Earth?
Gravity?QuickTime™ and a
TIFF (Uncompressed) decompressorare needed to see this picture.
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Electric fields? Lasers?
compilation by A.Ringwald
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Condensed matter black holes?
Unruh ‘81;e.g. T. Vachaspati, cond-mat/0404480“slow light”?
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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
Transition amplitude:
In dual resonance model:
Unitarity: P(mM)=const,
b=1/2 universal slope
limiting acceleration
M
Hagedorntemperature!
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Fermi - Landau statistical model of multi-particle production
Enrico Fermi1901-1954
Lev D. Landau1908-1968
Hadron productionat high energiesis driven bystatistical mechanics;universal temperature
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Where on Earth can one achieve the largestacceleration (deceleration) ?
Relativistic heavy ion collisions! -stronger color fields:
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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 stateschanges accordingly;this means that the average transverse momentum of partons should grow
CGC QGP
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Quantum thermal radiation at RHIC
The event horizon emerges due to the fastdecceleration of the colliding nucleiin strong color fields;
Tunneling throughthe event horizon leads to the thermalspectrum
Rindler and Minkowski spaces
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The emerging picture
Big question:
How does the producedmatter thermalize so fast?
Perturbation theory +Kinetic equations
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Fast thermalization
Rindler coordinates:
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
collision pointQs
horizons
Gluons tunneling through the event horizons have thermal distribution. They get on mass-shell in t=2Qs
(period of Euclidean motion)
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Deceleration-induced phase transitions?
• Consider Nambu-Jona-Lasinio model in Rindler space
e.g., Ohsaku ’04
€
L =ψ iγ ν (x)∇νψ (x) +λ
2N(ψ (x)ψ (x))2 + (ψ (x)iγ 5ψ (x))2
[ ]
• Commutation relations:
€
γμ (x),γν (x){ } = 2gμν (x)
€
2 = x 2 − t 2,
€
η =1
2ln
t + x
t − x
€
ds2 = ρ 2dη 2 − dρ 2 − dx⊥2
• Rindler space:
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Gap equation in an accelerated frame• Introduce the scalar and pseudo-scalar fields
€
σ(x) = −λ
Nψ (x)ψ (x)
€
(x) = −λ
Nψ (x)iγ 5ψ (x)
• Effective action (at large N):
€
Seff = d4 x −g −σ 2 + π 2
2λ
⎛
⎝ ⎜
⎞
⎠ ⎟− i lndet(iγ ν∇ν −σ − iγ 5π )∫
• Gap equation:
€
σ =−2iλσ
a
d2k
(2π )2∫ dωsinh(πω /a)
π 2(K iω / a +1/ 2(α /a))2 − (K iω / a−1/ 2(α /a))2){ }
−∞
∞
∫
where
€
α 2 = k⊥2 + σ 2
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Rapid deceleration induces phase transitions
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Nambu-Jona-Lasinio model(BCS - type)
Similar to phenomena in the vicinity of a large black hole: Rindler space Schwarzschild metric
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Black holes
A link between General Relativity and QCD?solution to some of the RHIC puzzles?
RHIC collisions