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