1. 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  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

2. 1.Introduction 1. Introduction Plasma = (partly) ionized gas (4. state of matter) 99% of the visible matter in universe Plasmas emit light What is a plasma?

3. 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

4. Lightening Aurora Flames Tubes Comets “Neon” Discharges Quantum Plasmas Relativistic Plasmas Sun Fusion Corona W. dwarfs Temperature Pressure 1 10 3 10 6 10 -3 10 -6 10 3 10 6 10 0 Kelvin Supernova bar Strongly coupled Plasmas ComplexPlasmas

5. What is an electron-positron plasma? Strong electric or magnetic fields, high temperatures  massive pair production (E > 2m e c 2 = 1.022 MeV)  electron-positron plasma Astrophysical examples: Supernovae: T max = 3 x 10 11 K  kT = 30 MeV >> 2m e c 2 Magnetars: Neutron Stars with strong magnetic fields B > 10 14 G Accretion disks around Black Holes

6. High-intensity lasers (I > 10 24 W/cm 2 ) ELI: The Extreme Light Infrastructure European Project Recent developments in laser technology  ultrashort pulses (10 -18 s), ultrahigh intensities (> 10 23 W/cm 2 )  observation of ultra-fast processes (molecules), particle acceleration, ultradense matter, electron-positron plasma

7. 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. 10 14 V/cm corresponding to 5 x 10 25 W/cm 2 (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.

8. Here: Properties of a thermalized electron-positron plasma, not production and equlibration Equation of state Assumptions: ultrarelativistic gas: T >> m e (  = c = k =1) thermal and chemical equilibrium electron density = positron density  zero chemical potential ideal gas (no interactions) infinitely extended, homogeneous and isotropic Electron and positron distribution function: Photon distribution function: Ultrarelativistic particles: E = p Particle number density: 2. Field Theoretic description of Electron-Positron Plasmas

9. Example: T = 10 MeV  Conversion: 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

10. Volume of neutron star (10 km diameter)  E ~ 10 41 J corresponding to about 10% of entire Supernova energy (without neutrinos) Volume 1  m 3  E = 3.8 x 10 11 J = 0.1 kto TNT Energy of a laser pulse about 100 J at I > 10 24 W/cm 2 ! Is the ideal gas approximation reliable? Coulomb coupling parameter:  C = e 2 /(dT) Interparticle distance: d ~ (  eq e ) -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?

11. 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, m e  Debye screening length  Plasma frequency Ultrarelativistic plasmas: scales T (hard momenta), eT (soft momenta) Collective phenomena: soft momenta Transport properties: hard momenta

12. Relativistic interactions between electrons  QED Perturbation theory: Expansion in  = e 2 /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)

14. 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:

15. 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)

16. Maxwell equations   propagation of collective plasma modes  dispersion relations Plasma frequency: Yukawa potential: with Debye screening length Landau damping  pl Plasmon

17. Relativistic plasmas  Fermionic plasma modes: dispersion relation of electrons and positrons in plasma Electron self-energy:  electron dispersion relation (pole of effective electron propagator containing electron self-energy)  Plasmino branch Note: minimum in plasmino dispersion  van Hove singularity  unique opportunity to detect fermionic modes in laser produced plasmas modes in laser produced plasmas

18. 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  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

19. 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

20. Photon Production Thermal distribution of electrons and positrons, expansion of plasma droplet (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

21. EoS Collective Transport

22. Chemical non-equilibrium T= 10 MeV  equilibrium electron-positron number density Experiment: colliding laser pulses  electromagn. cascade, laser depletion  max. electron-positron number about 10 13 in a volume of about 0.1  m 3 (diffractive limit of laser focus) at I = 2.7 x 10 26 W/cm 2  (A.M. Fedotov et al., PRL 105 (2010) 080402)  exp <  eq  non-equilibrium plasma Assumption: thermal equilibrium but no chemical equilibrium  electron distribution function f F =  n F with fugacity  2

23. 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 

24. 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

25. 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 