quantum field theoretic description of electron-positron plasmas markus h. thoma...
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Quantum Field Theoretic Description of Quantum Field Theoretic Description of
Electron-Positron Plasmas Electron-Positron Plasmas
Markus H. ThomaMax-Planck-Institute for Extraterrestrial Physics, Univ. Giessen, MAP,
EMMI, Berner & Mattner Systemtechnik
Ultrastrong laser, supernovae electron-positron plasma prediction of properties necessary 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
1.1. IntroductionIntroduction
Plasma = (partly) ionized gas (4. state of matter) 99% of the visible matter in universePlasmas emit light
What is a plasma?
Plasmas can be produced by high temperatures electric fields radiation
Relativistic plasmas: (Supernovae)
Quantum plasmas: (White Dwarfs)
Strongly coupled plasmas: (WDM, Dusty Plasmas, QGP)
C: Coulomb coupling parameter = Coulomb energy / thermal energy
Bth
hd
m v
2kT mc
2
C
Q1
d kT
Lightening
Aurora
Flames
TubesTubes
Comets
“Neon”“Neon”
Discharges
Quantum Plasmas
Rel
ativ
istic
Pla
smasSun
Fusion
Corona
W. dwarfs
Temperature
Pre
ssur
e
1
103
106
10-3
10-6
103 106100 Kelvin
Supernovabar
Strongly coupled Plasmas
Complex Complex PlasmasPlasmas
What is an electron-positron plasma?
Strong electric or magnetic fields, high temperatures massive pair production (E > 2mec2 = 1.022 MeV) electron-positron plasma
Astrophysical examples:
• Supernovae: Tmax = 3 x 1011 K 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
ultrashort pulses (10-18 s), ultrahigh intensities (> 1023 W/cm2)
observation of ultra-fast processes (molecules), particle acceleration,ultradense matter, electron-positron plasma
Possibilities for electron-positron plasma formation:
• Thin gold foil (~1 m) hit by two laser pulses from opposite sides (B. Shen, J. Meyer-ter-Vehn, Phys. Rev. E 65 (2001) 016405) target electrons heated up to multi- MeV temperatures e- - e+ plasma
• Colliding laser pulses pair creation at about 1/100 of critical field strength, i.e. 1014 V/cm corresponding to 5 x 1025 W/cm2 (ELI, XFEL) 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)
Habs et al.
Here: Properties of a thermalized electron-positron plasma, not productionand equlibration
• Equation of state
Assumptions:• ultrarelativistic gas: T >> me ( = c = k =1)• thermal and chemical equilibrium• electron density = positron density zero chemical potential• ideal gas (no interactions)• infinitely extended, homogeneous and isotropic
B E / Tn
e
1
1
F E / Tn
e
1
1
eqe F F F
d pg n ( p) . T , g
( )
3
33
0 37 42
Electron and positron distribution function:
Photon distribution function:
Ultrarelativistic particles: E = p
Particle number density:
2. Field Theoretic description of Electron-Positron Plasmas2. Field Theoretic description of Electron-Positron Plasmas
Example: T = 10 MeV Conversion:
eqe MeV 3370
c . MeV m MeV . J . m . s13 13 12 1 21 11 1 97 10 1 1 60 10 5 08 10 1 52 10
eqe . m40 34 9 10
eqF F B B
d p d pg pn ( p) g pn ( p) . T
( ) ( )
3 3
43 3
1 812 2
eqB B B
d pg n ( p) . T , g
( ) 3
33
0 24 22
eq . J m29 33 8 10
Photon density: Photons in equilibrium with electrons and positrons electron-positron-photon gas
Energy density: Stefan-Boltzmann law
T = 10 MeV:
Photons contribute 36% to energy density
Volume of neutron star (10 km diameter) E ~ 1041 J corresponding to about 10% of entire Supernova energy (without neutrinos)
Volume 1 m3 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: C = e2/(dT)
Interparticle distance: d ~ (eqe)-1/3 = 2.7 x 10-14 m at T = 10 MeV
C = 5.3 x 10-3
weakly coupled QED plasma
equation of state of an ideal gas is a good approximation; interactions can be treated by perturbation theory
Quark-gluon plasma: C = 1 – 5 quark-gluon plasma liquid?
• Collective phenomena
Interactions between electrons and positrons 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
Debye screening length
Plasma frequency
Ultrarelativistic plasmas: scales T (hard momenta), eT (soft momenta)
Collective phenomena: soft momenta Transport properties: hard momenta
eD e
kT
24
e
epl m
e 24
Relativistic interactions between electrons QED
Perturbation theory: Expansion in = e2/4 =1/137 (e = 0.3) using Feynman diagrams, e.g. electron-electron scattering
Evaluation of diagrams by Feynman rules 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 2 independent components depending on frequency and momentum k=|k|
High-temperature or hard thermal loop limit (T >> , k ~ eT):
Effective photon mass:
L
T
km ln ,
k k
k km ln
k k k
2
2 22
2 2
3 12
31 1
2 2
eTm
3T MeV m MeV 10 1
Dielectric tensor:
Momentum space:
Isotropic medium:
Relation to polarization tensor:
Alternative derivation using transport theory (Vlasov + Maxwell equations)
Same result for quark-gluon plasma (apart from color factors)
LL
TT
m( ,k ) k( ,k ) ln
k k k k
m( ,k ) k k( ,k ) ln
k k k
2
2 2
2 2
2 2 2
31 1 1
2
31 1 1 1
2 2
i ij jD ( ,k ) ( ,k )E ( ,k ) i , j x,y ,z
i j i jij T ij L
k k k k( ,k ) ( ,k ) ( ,k )
k k
2 2
Maxwell equations
propagation of collective plasma modes dispersion relations
L T
k( ,k ) , ( ,k )
2
20
Plasma frequency:
Yukawa potential:
with Debye screening length
pl L,T (k ) m
. Hz (T MeV )
21
0
1 5 10 10
Landau dampingpl
DreV(r )
r
Dm
. m (T MeV )
13
1
3
1 1 10 10
L,Tk Im 2 2 0
Plasmon
Relativistic plasmas Relativistic plasmas Fermionic plasma modesFermionic plasma modes: : dispersion relation of electrons and positrons in plasma dispersion relation of electrons and positrons in plasma
Electron self-energy: Electron self-energy:
electron dispersion relation (poleelectron dispersion relation (poleof effective electron propagator containingof effective electron propagator containingelectron self-energy)electron self-energy)
PlasminoPlasmino branch branch
Note: minimum in plasmino dispersionNote: minimum in plasmino dispersion
van Hove singularityvan Hove singularity
unique opportunity to detect fermionicunique opportunity to detect fermionic modes in laser produced plasmasmodes 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 electron damping (interaction) rate, electon energy loss, shear viscosity k
Problem: IR divergence
222
* ,1
Tek
D LL
L
HTL perturbation theory (Braaten, Pisarski, Nucl. Phys. B337 (1990) 569)
Resummed photon propagator for soft photon momenta, i.e. k ~ eT
IR improved (Debye screening), gauge independent results complete to leading order
• Electron damping rates and energy loss
• Transport coefficients of e--e+ plasma, e.g. shear viscosity
• Photon damping
Mean free path 1/ph = 0.3 nm for T=10 MeV for a thermal photon
ev
Tee
1ln
4
2
)1ln(4
3
ee
T
Te
E
E
Teph 2
24
ln64
• Photon Production
Thermal distribution of electrons and positrons, expansion of plasmadroplet (hydrodynamical model)
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
T= 10 MeV equilibrium electron-positron number density
Experiment: colliding laser pulses electromagn. cascade, laser depletion
max. electron-positron number about 1013 in a volume of about 0.1 m3 (diffractive limit of laser focus) at I = 2.7 x 1026 W/cm2
(A.M. Fedotov et al., PRL 105 (2010) 080402)
exp< eq non-equilibrium plasma
Assumption: thermal equilibrium but no chemical equilibrium
electron distribution function fF = nF with fugacity
eq m40 310
exp m32 310
exp F F eq
d pg n ( p)
( )
38
310
2
2
Non-equilibrium QED:
M.E. Carrington, H. Defu, M.H. Thoma, Eur. Phys. C7 (1999) 347
Electron plasma frequency in sun (center):
Debye screening length:
Collective effects important since extension of plasma L ~ 1 m >> D
Electron density > positron density finite chemical potential
F
em dp pf ( p)
22
20
4
3
eq
pl
m m eV
. Hz17
100
1 5 10
pl Hz 175 10
D nm1
• Particle production
Temperature high enough new particles are produced
Example: Muon production via
Equilibrium production rate:
Invariant photon mass:
Muon production exponentially suppressed at low temperatures T < m= 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
( E p ) T
E T ( E p ) T
m mdN T eln
d xd p M M p e e
2 22 2
4 4 4 2 2 2
2 4 1 11 1
36 1 1
M E p m2 2 2 24
3. Summary
• Aim: prediction of properties of ultrarelativistic electron-positron plasmas produced in laser fields and supernovae
• Ultrarelativistic electron-positron plasma: weakly coupled system ideal gas equation of state (in contrast to QGP)
• Interactions in plasma perturbative QED at finite temperature 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 perturbative QED in non-equilibrium