Quantum Black Holes, Strong Fields, and Relativistic Heavy Ions
D. Kharzeev
“Understanding confinement”, May 16-21, 2005
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
Black holes emitthermal radiationwith temperature
S.Hawking ‘74
acceleration of gravityat 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 detectsa 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
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 tunnelingunder 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
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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
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
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
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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
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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
Pair production by a pulse
€
Aμ = 0,0,0,−E
k0
tanh(k0t) ⎛
⎝ ⎜
⎞
⎠ ⎟
Consider a time dependent field E
t
• Constant field limit
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k0 → 0
• Short pulse limit
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k0 → ∞
a thermal spectrum with
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
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:
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
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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!
Fermi - Landau statistical model of multi-particle production
Enrico Fermi1901-1954
Lev D. Landau1908-1968
Hadron productionat high energiesis driven bystatistical mechanics;universal temperature
Where on Earth can one achieve the largestacceleration (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 stateschanges 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 fastdecceleration of the colliding nucleiin strong color fields;
Tunneling throughthe event horizon leads to the thermalspectrum
Rindler and Minkowski spaces
The emerging picture
Big question:
How does the producedmatter thermalize so fast?
Perturbation theory +Kinetic equations
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)
Deceleration-induced phase transitions?
• Consider Nambu-Jona-Lasinio model in Rindler space
e.g., Ohsaku ’04
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L =ψ iγ ν (x)∇νψ (x) +λ
2N(ψ (x)ψ (x))2 + (ψ (x)iγ 5ψ (x))2
[ ]
• Commutation relations:
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γμ (x),γν (x){ } = 2gμν (x)
€
2 = x 2 − t 2,
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η =1
2ln
t + x
t − x
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ds2 = ρ 2dη 2 − dρ 2 − dx⊥2
• Rindler space:
Gap equation in an accelerated frame• Introduce the scalar and pseudo-scalar fields
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σ(x) = −λ
Nψ (x)ψ (x)
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(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
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α 2 = k⊥2 + σ 2
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