Quantum Field Theoretic Description of
Electron-Positron Plasmas
Markus H. Thoma
Max-Planck-Institute for Extraterrestrial Physics, Univ. Giessen, MAP,
EMMI, Berner & Mattner Systemtechnik
Ultrastrong laser, supernovae
g electron-positron plasma
g prediction of properties necessary
g quantum field theoretic methods developed mainly for quark-gluon plasma
1.
Introduction
2.
Field Theoretic Description of Electron-Positron Plasmas
3. Summary
M.H. Thoma, arXiv:0801.0956, Rev. Mod. Phys. 81 (2009) 959
1. Introduction
What is a plasma?
Plasma = (partly) ionized gas (4. state of matter)
99% of the visible matter in universe
Plasmas emit light
Plasmas can be produced by
high temperatures
electric fields
kT  mc2
Relativistic plasmas:
Quantum plasmas:
B 
Strongly coupled plasmas:
h
d
m v th
Q2
GC 
1
d kT
radiation
(Supernovae)
(White Dwarfs)
(WDM, Dusty Plasmas, QGP)
GC: Coulomb coupling parameter = Coulomb energy / thermal energy
W. dwarfs
Supernova
Quantum Plasmas
bar
106
Pressure
103
Relativistic Plasmas
Complex
Plasmas
Sun
Strongly coupled
Plasmas
Flames
1
Lightening
“Neon”
Tubes
Fusion
10-3
Discharges
10-6
Aurora
Corona
Comets
100
103
Temperature
106
Kelvin
What is an electron-positron plasma?
Strong electric or magnetic fields, high temperatures
g massive pair production (E > 2mec2 = 1.022 MeV)
g electron-positron plasma
Astrophysical examples:
• Supernovae: Tmax = 3 x 1011 K g kT = 30 MeV >> 2mec2
• Magnetars: Neutron Stars with strong magnetic fields B > 1014 G
• Accretion disks around Black Holes
High-intensity lasers (I > 1024 W/cm2)
ELI: The Extreme Light Infrastructure European Project
Recent developments in laser technology
gultrashort pulses (10-18 s), ultrahigh intensities (> 1023 W/cm2)
g observation of ultra-fast processes (molecules), particle acceleration,
ultradense matter, electron-positron plasma
Possibilities for electron-positron
plasma formation:
Habs et al.
• Thin gold foil (~1 mm) hit by two
laser pulses from opposite sides
(B. Shen, J. Meyer-ter-Vehn,
Phys. Rev. E 65 (2001) 016405)
g target electrons heated up to multiMeV temperatures g e- - e+ plasma
• Colliding laser pulses g pair creation at about 1/100 of critical field
strength, i.e. 1014 V/cm corresponding to 5 x 1025 W/cm2 (ELI, XFEL)
g electromagnetic cascade, depletion of laser energy
(A.M. Fedotov et al., PRL 105 (2010) 080402)
• Laser-electron beam interaction (ELI-NP: two 10 PW lasers plus
600 MeV electron beam) (D. Habs, private communication)
2. Field Theoretic description of Electron-Positron Plasmas
Here: Properties of a thermalized electron-positron plasma, not production
and equlibration
• Equation of state
Assumptions:
• ultrarelativistic gas: T >> me (h = c = k =1)
• thermal and chemical equilibrium
• electron density = positron density g zero chemical potential
• ideal gas (no interactions)
• infinitely extended, homogeneous and isotropic
Electron and positron distribution function:
1
n

B
Photon distribution function:
eE / T  1
nF 
1
eE / T  1
Ultrarelativistic particles: E = p
Particle number density:

eq
e
d3p
3
 gF 
n
(
p)

0
.
37
T
, gF  4
F
3
( 2 )
3

370
MeV
Example: T = 10 MeV g eq
e
Conversion:
1  c  1.97  1013 MeV m  1MeV  1.60  1013 J  5.08  1012 m1  1.52  1021s 1
40
3
 eq
e  4.9  10 m
Photon density: Photons in equilibrium with electrons and positrons
g electron-positron-photon gas

eq

d3p
3
 gB 
n
(
p)

0
.
24
T
, gB  2
B
3
( 2 )
Energy density: Stefan-Boltzmann law

eq
d3p
d 3p
4
 gF 
p
n
(
p)

g
p
n
(
p)

1
.
81
T
F
B
B
 ( 2 )3
( 2 )3
T = 10 MeV:
eq  3.8  1029 J m3
Photons contribute 36% to energy density
Volume of neutron star (10 km diameter) g E ~ 1041 J corresponding
to about 10% of entire Supernova energy (without neutrinos)
Volume 1 mm3 g E = 3.8 x 1011 J = 0.1 kto TNT
Energy of a laser pulse about 100 J at I > 1024 W/cm2 !
Is the ideal gas approximation reliable?
Coulomb coupling parameter: GC = e2/(dT)
Interparticle distance: d ~ (eqe)-1/3 = 2.7 x 10-14 m at T = 10 MeV
gGC = 5.3 x 10-3
g weakly coupled QED plasma
g equation of state of an ideal gas is a good approximation;
interactions can be treated by perturbation theory
Quark-gluon plasma: GC = 1 – 5 g quark-gluon plasma liquid?
• Collective phenomena
Interactions between electrons and positrons g collective phenomena,
e.g. Debye screening, plasma waves, transport properties, e.g. viscosity
Non-relativistic plasmas (ion-electron):
Classical transport theory with scales: T, me
g Debye screening length
Plasma frequency  pl 
D 
kT
4 e 2  e
4 e 2  e
me
Ultrarelativistic plasmas: scales T (hard momenta), eT (soft momenta)
Collective phenomena: soft momenta
Transport properties: hard momenta
Relativistic interactions between electrons g QED
Perturbation theory: Expansion in a = e2/4 =1/137 (e = 0.3)
using Feynman diagrams, e.g. electron-electron scattering
Evaluation of diagrams by Feynman rules g
scattering cross sections, damping and production rates, life times etc.
Interactions within plasma: QED at finite temperature
Extension of Feynman rules to finite temperature (imaginary or real
time formalism), calculations more complicated than at T=0
Application: quark-gluon plasma (thermal QCD)
Example: Photon self-energy or polarization tensor (K=(,k))
Isotropic medium g 2 independent components depending on
frequency  and momentum k=|k|
High-temperature or hard thermal loop limit (T >> , k ~ eT):
  k 

 L  3m2  1 
ln
,

2k   k 

3 2 2
T  m 2
2
k
 
k2     k 
ln
1   1  2 


2
k


k

 

Effective photon mass:
m 
eT
3
T  10 MeV  m  1MeV
Dielectric tensor:
Momentum space:
Di ( ,k )  ij ( ,k )E j ( ,k )
Isotropic medium:
ki k j

ij ( ,k )  T ( ,k )  ij  2
k

i, j  x,y,z
ki k j

  L ( ,k ) 2
k

Relation to polarization tensor:
3m
 L ( ,k )
L ( ,k )  1 
 1 2
2
k
k
  k 

1

ln

2k   k 

3m
T ( ,k )
T ( ,k )  1 
 1
2

2k 2
 
k2     k 
ln
1   1  2 


2
k


k

 

2
2
Alternative derivation using transport theory (Vlasov + Maxwell equations)
Same result for quark-gluon plasma (apart from color factors)
k2
Maxwell equations g L ( ,k )  0, T ( ,k )  2

g propagation of collective plasma modes
g dispersion relations
Plasma frequency:
pl  L,T ( k  0 )  m
 1.5  1021 Hz (T  10 MeV )
Plasmon
pl
Landau damping
2  k 2  Im L,T  0
Yukawa potential:
e  r D
V( r ) 
r
with Debye screening length
D 
1
3 m
 1.1 10 13 m (T  10 MeV )
Relativistic plasmas g Fermionic plasma modes:
dispersion relation of electrons and positrons in plasma
Electron self-energy:
gelectron dispersion relation (pole
of effective electron propagator containing
electron self-energy)
g Plasmino branch
Note: minimum in plasmino dispersion
gvan Hove singularity
g unique opportunity to detect fermionic
modes in laser produced plasmas
• Transport properties
Transport properties of particles with hard (thermal) momenta (p ~ T) using
perturbative QED at finite temperature
p~T
For example electron-electron scattering
g electron damping (interaction) rate,
electon energy loss, shear viscosity
k
Problem: IR divergence
gHTL perturbation theory (Braaten, Pisarski, Nucl. Phys. B337 (1990) 569)
Resummed photon propagator for soft photon momenta, i.e. k ~ eT
1
D  2
,  L  e2T 2
k  L
*
L
g IR improved (Debye screening), gauge independent results complete to
leading order
• Electron damping rates and energy loss
e2T
1
e 
v ln
4
e
• Transport coefficients of e--e+ plasma, e.g. shear viscosity
T3
 4
e ln(1 e)
• Photon damping
 ph
e4T 2
E

ln 2
64 E e T
Mean free path 1/ph = 0.3 nm for T=10 MeV for a thermal photon
• Photon Production
Thermal distribution of electrons and positrons, expansion of plasma
droplet (hydrodynamical model)
g Gamma ray flash from plasma droplet shows continuous spectrum
(not only 511 keV line)
M.G. Mustafa, B. Kämpfer,
Phys. Rev. A 79 (2009) 020103
EoS
Collective
Transport
• Chemical non-equilibrium
40
3
T= 10 MeV g equilibrium electron-positron number density eq  10 m
Experiment: colliding laser pulses g electromagn. cascade, laser depletion
gmax. electron-positron number about 1013 in a volume of about 0.1 mm3
(diffractive limit of laser focus) at I = 2.7 x 1026 W/cm2
32
3
g exp  10 m
2
(A.M. Fedotov et al., PRL 105 (2010) 080402)
exp< eq g non-equilibrium plasma
Assumption: thermal equilibrium but no chemical equilibrium
g electron distribution function fF =  nF with fugacity   1
exp
d 3p
8
 gF 

n
(
p)





10
F
eq
( 2 )3

Non-equilibrium QED:
2
4
e
m2 
dp p fF ( p )
2 
3 0
M.E. Carrington, H. Defu, M.H. Thoma, Eur. Phys. C7 (1999) 347
 m   meq  100 eV
 pl  1.5  1017 Hz
Electron plasma frequency in sun (center): pl  5  10 Hz
17
Debye screening length:
D  1nm
Collective effects important since extension of plasma L ~ 1 mm >> D
Electron density > positron density g finite chemical potential m
• Particle production
Temperature high enough g new particles are produced
Example: Muon production via
Equilibrium production rate:
2
2
 1  e ( E  p ) 2T 
dN
a 2  2mm  4mm  T
1

1  2  1  2 
ln 

4
4
4 
ET



d xd p 36 
M 
M  p e  1  1  e ( E  p ) 2T 
Invariant photon mass: M 2  E2  p2  4mm2
Muon production exponentially suppressed at low temperatures
T < mm= 106 MeV
Very high temperatures (T > 100 MeV):
Hadronproduction (pions etc.) and Quark-Gluon Plasma
I. Kuznetsova, D. Habs, J. Rafelski, Phys. Rev. D 78 (2008) 014027
3. Summary
• Aim: prediction of properties of ultrarelativistic electron-positron
plasmas produced in laser fields and supernovae
• Ultrarelativistic electron-positron plasma: weakly coupled system
g ideal gas equation of state (in contrast to QGP)
• Interactions in plasma g perturbative QED at finite temperature
g collective phenomena (plasma waves, Debye screening) and
transport properties (damping rates, mean free paths, relaxation times,
production rates, viscosity, energy loss) using HTL resummation
• New phenomenon: Fermionic collective plasma modes (plasmino),
van Hove singularities?
• Deviation from chemical equilibrium g perturbative QED in
non-equilibrium