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: , Rev. Mod. Phys. 81 (2009) 959

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?

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

Lightening Aurora Flames Tubes Comets “Neon” Discharges Quantum Plasmas Relativistic Plasmas Sun Fusion Corona W. dwarfs Temperature Pressure Kelvin Supernova bar Strongly coupled Plasmas ComplexPlasmas

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

High-intensity lasers (I > W/cm 2 ) ELI: The Extreme Light Infrastructure European Project Recent developments in laser technology  ultrashort pulses ( s), ultrahigh intensities (> W/cm 2 )  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) )  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 V/cm corresponding to 5 x W/cm 2 (ELI, XFEL)  electromagnetic cascade, depletion of laser energy (A.M. Fedotov et al., PRL 105 (2010) ) 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 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

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

Volume of neutron star (10 km diameter)  E ~ J corresponding to about 10% of entire Supernova energy (without neutrinos) Volume 1  m 3  E = 3.8 x J = 0.1 kto TNT Energy of a laser pulse about 100 J at I > 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 m at T = 10 MeV   C = 5.3 x   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, m e  Debye screening length  Plasma frequency Ultrarelativistic plasmas: scales T (hard momenta), eT (soft momenta) Collective phenomena: soft momenta Transport properties: hard momenta

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)

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:

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)

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

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

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

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

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)

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 in a volume of about 0.1  m 3 (diffractive limit of laser focus) at I = 2.7 x W/cm 2  (A.M. Fedotov et al., PRL 105 (2010) )  exp <  eq  non-equilibrium plasma Assumption: thermal equilibrium but no chemical equilibrium  electron distribution function f F =  n F with fugacity  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 

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)

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 