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Quantum Field Theoretic Description of Electron-Positron Plasmas

Learn about the properties and formation of electron-positron plasmas using quantum field theoretic methods. Explore astrophysical examples, interactions, and collective phenomena in this cutting-edge field of research.

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Quantum Field Theoretic Description of Electron-Positron Plasmas

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  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 • 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 • Summary M.H. Thoma, arXiv:0801.0956, Rev. Mod. Phys. 81 (2009) 959

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

  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) GC: Coulomb coupling parameter = Coulomb energy / thermal energy

  4. Quantum Plasmas Supernova W. dwarfs 106 Sun 103 Flames Lightening Tubes Pressure “Neon” 1 Relativistic Plasmas Fusion 10-3 Discharges Aurora Corona 10-6 Comets 100 103 106 Kelvin Temperature bar Complex Plasmas Strongly coupled Plasmas

  5. 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 g kT = 30 MeV >> 2mec2 • Magnetars: Neutron Stars with strong magnetic fields B > 1014 G • Accretion disks around Black Holes

  6. 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) • g observation of ultra-fast processes (molecules), particle acceleration, • ultradense matter, electron-positron plasma

  7. Possibilities for electron-positron • plasma formation: • 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 multi- • MeV 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) Habs et al.

  8. 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: Photon distribution function: Ultrarelativistic particles: E = p Particle number density:

  9. Example: T = 10 MeVg Conversion: Photon density: Photons in equilibrium with electrons and positrons g 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) gE ~ 1041 J corresponding • to about 10% of entire Supernova energy (without neutrinos) • Volume 1 mm3g 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 ~ (reqe)-1/3 = 2.7 x 10-14 m at T = 10 MeV • GC = 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: GC = 1 – 5 gquark-gluon plasma liquid?

  11. 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 • 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 gQED Perturbation theory: Expansion in a = e2/4p =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)

  13. Example: Photon self-energy or polarization tensor (K=(w,k)) Isotropic medium g 2 independent components depending on frequency w and momentum k=|k| High-temperature or hard thermal loop limit (T >> w, k ~ eT): Effective photon mass:

  14. 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)

  15. Maxwell equations g • propagation of collective plasma modes • dispersion relations Plasma frequency: Yukawa potential: with Debye screening length Plasmon Landau damping wpl

  16. Relativistic plasmas gFermionic 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

  17. 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 • HTL perturbation theory (Braaten, Pisarski, Nucl. Phys. B337 (1990) 569) Resummed photon propagator for soft photon momenta, i.e. k ~ eT g IR improved (Debye screening), gauge independent results complete to leading order

  18. Electron damping rates and energy loss • Transport coefficients of e--e+ plasma, e.g. shear viscosity • Photon damping • Mean free path 1/gph = 0.3 nm for T=10 MeV for a thermal photon

  19. 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

  20. EoS Collective Transport

  21. Chemical non-equilibrium • T= 10 MeVg equilibrium electron-positron number density • Experiment: colliding laser pulses g electromagn. cascade, laser depletion • max. 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 (A.M. Fedotov et al., PRL 105 (2010) 080402) • rexp< reqg non-equilibrium plasma • Assumption: thermal equilibrium but no chemical equilibrium • electron distribution function fF = l nFwith fugacity l < 1 2

  22. 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 mm >> lD Electron density > positron density g finite chemical potential m

  23. Particle production • Temperature high enough g new particles are produced • Example: Muon production via • Equilibrium production rate: • Invariant photon mass: • 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

  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 • 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

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