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INTRODUCTION TO STRONG FIELD QED
TOM HEINZLGK SEMINAR, TPI JENAGK SEMINAR, TPI JENA21/04/2009
with: C. Harvey (UoP), A. Ilderton (Dublin), K. Ledingham (Strathclyde), H. Schwoerer (Stellenbosch), R. Sauerbrey and U. Schramm (FZD), A. Wipf (FSU Jena)
Outline
1. Introduction
2. Strong Field Electrodynamics
Application I: Relativistic charges in laser fields
3. Quantum Electrodynamics3. Quantum Electrodynamics
4. Strong Field QED
Application II: vacuum polarisation effects
Application III: nonlinear Compton scattering [tomorrow]
5. Summary and Conclusions
1. Introduction
Explaining the title
� “QED” = quantum electrodynamics
� Quantum gauge field theory
� describes interactions of “light” and “matter”
� “light”: photons� “light”: photons
� “matter”: charged elementary particles
� Leptons: electron, muon, tau
� Quarks: up, down, strange, … (bound into hadrons)
� NB1: mainly discuss electrons
� NB2: “matter” includes anti-matter (positrons etc.)
Explaining the title contd
� “Strong Field”
� Throughout this lecture: strong field = laser
� Typical magnitudes assumed are
Power
Intensity
Electric field
Magnetic field
Strong field parameter
� ‘dimensionless laser amplitude’
� (purely classical) ratio (no ):� (purely classical) ratio (no ):
� NB: implies relativistic quiver motion of electrons in laser beam
Relativistic charges in electromagnetic fields
2. Strong Field Electrodynamics
Motivation
� Direct laser acceleration (DLA) in vacuum –
� Usual RF cavities break down at ‘critical’ electric field of about 108 V/m
� Can one use a laser to accelerate particles?
� No – in a perfect plane wave (Lawson-Woodward theorem)
� Yes – in ‘realistic’ laser fields
� experiments at proof-of-principle stage
� 30 → 30.03 MeV (Plettner, PRL 95, 134801 (2005))
� 40 → 43.7 MeV (Campbell et al., IEEE TPS 28, 1094 (2000))
Covariant equation of motion
� Charge e, mass m in field
� Equation of motion:
where dot denotes derivative w.r.t. proper time
� R.h.s.: relativistic Lorentz force
� Task: find trajectory
� NB: as EoM in general nonlinear!
Simplification I: constant fields
� EoM becomes linear for constant fields,
� Write in matrix form
where
� First integral :
� Second integral: 4 cases – depending on invariants
Constant fields: 4 cases (Taub 1948)
� Table:
Name Field configuration
(special frame)
Invariant characterisation
(frame independent)
Loxodromic
� NB: parabolic = crossed or null fields,
Loxodromic
Elliptic
Hyperbolic
Parabolic
4 cases: illustration
� NB: net acceleration
� Hyperbolic: basic principle of standard accelerator
Loxodromic:
Hyperbolic Elliptic
� Loxodromic: accelerator with magnetic focussing
� Parabolic: laser subcycle acceleration
� Elliptic: no acceleration
Loxodromic Parabolic
(C. Harvey)
Simplification II: plane waves
� Covariant description of plane wave field
� Light-like wave 4-vector , →
� dispersion for massless particles
� Ingredients � Ingredients
� Invariant phase
� Field strength:
� Transversality:
� Null field:
� Conservation law:
makes EoM linear, hence integrable
Illustration I: linear polarisation
� Trajectory of charge in average rest frame
(mean drift velocity subtracted)
Lissajous 2:1
Illustration II: Circular polarisation
� Trajectory of charge in average rest frame
(mean drift velocity subtracted)
DLA: Conclusion
� Motion in periodic plane wave fields ( )
� Periodic and bounded
� Hence no net acceleration (Lawson-Woodward theorem)
� Loophole: give up periodicity, e.g. crossed fields ( )
� Motion aperiodic and unbounded
� Net (‘subcycle’) acceleration
DLA: Outlook
� Work in progress
� realistic laser fields
� include radiation loss
� numerical scheme� numerical scheme
� Application
� Plug calculated orbits into Larmor formula
� Determine classical radiation spectrum(Sarachik/Schappert 1970, Esarey et al. 1996, ...)
3.1 Introduction
3. QED
Switching on h-bar
� Recall dimensionless laser amplitude: energy gain across laser wavelength in units of
for relativistic electronsfor relativistic electrons
� Replace laser by Compton wavelength
and demand energy gain across to be
Critical electric field
� This yields ‘critical’ electric field (Sauter 1931)
� In this field a change in energy occurs within � In this field a change in energy occurs within microscopic length scale
� Hence, it becomes possible to create electron positron pairs from vacuum
� Presence of and c: relativity ∪∪∪∪ quantum theory
� Need relativistic quantum field theory: QED!
QED Lagrangian
� Compact version:
� Covariant derivative� Covariant derivative
� guarantees gauge invariance under
→ photon massless
� determines interaction
QED: basic interaction
� Rewrite interaction term
� So, coupling of gauge field to Dirac current
� Pictorially: Feynman diagram of ‘QED vertex’
Photon field
Dirac field
Dirac field
coupling strength
Feynman diagrams
� Tree level, e.g. � Loops, e.g.
� Compton scattering
� O( )
� Typically finite
[tomorrow’s talk]
� Vacuum polarisation
� O( )
� Typically infinite
� If so, renormalise
Photon-photon scattering
� Loop effects imply photon-photon coupling mediated by virtual Dirac particles
� Feynman diagram (finite due to gauge invariance!)� Feynman diagram (finite due to gauge invariance!)
� Induced nonlinearity: effective terms in Lagrangian, in EoM
(Low energy: Euler, Heisenberg, Kockel 1935, 1936High energy: Akhiezer 1937)
Photon-photon scattering contd
� Low energy analysis:
� At low energy: virtual loop not resolved
� Obtain effective theory with effective vertices
� Heisenberg-Euler effective Lagrangian: nonlinear quantum correction to Maxwell theory
Heisenberg-Euler
� Discussion:
� from 4 QED vertices
� from dimensional analysis (Lagrangian has mass � from dimensional analysis (Lagrangian has mass dimension 4 in d=4)
� and from gauge invariance
recall: ,
� Coefficients and from detailed calculation
Photon-photon scattering contd
� Low-energy cross section
� Invariant amplitude: each gradient from etc. produces frequency factor
2
� X-section (optical regime):
� NB: not measured yet (→ bounds)
suggestion for Astra Gemini: (Marklund et al., 2006)
2
Loops vs. trees: optical theorem
� Optical theorem (Kramers-Kronig):
“the imaginary part of the forward scattering amplitude is proportional to the total cross section”
� Example: photon-photon scattering � Example: photon-photon scattering
Pair production cross section → absorption
Scattering loop
Vacuum polarisation effects
4. Strong Field QED
Strong field QED
� Recall QED interaction
� Assume presence of external strong laser field
� Get additional interaction (no )
� Include into free Lagrangian� Include into free Lagrangian
� Main effect: replace free Dirac electrons by Volkov
electrons (electrons ‘dressed’ by e.m. wave)
� Pictorially: ‘dressed’ (Volkov) electron line
Field decomposition
� Two types of photons
� Weak ‘probe field’ : perturbative
� Strong background field : nonperturbative
� Effective Lagrangian = expansion in� Effective Lagrangian = expansion in
� Goal: determine ‘coefficients’ exactly
� Only possible for special backgrounds
� Consider leading (bilinear) order
� coefficient = dressed vacuum polarisation loop:
Strong field vacuum polarisation
� coefficient = dressed vacuum polarisation loop:
LO SFQED
Polarisation tensor QED
Polarisation tensor: crossed fields
� Simplest case: crossed fields (constant null fields)
� Consequence: laser background has
� Two remaining invariants: � Two remaining invariants:
� Probe 4-momentum squared:
where is index of refraction due to laser BG
� Energy density seen by probe:
BG energy-momentum tensor
crossed fields contd
� Determine relevant eigenvalues of
� : two nontrivial dispersion relations
� NB: can be viewed as stemming from effective � NB: can be viewed as stemming from effective metrics
� Result:
‘vacuum’ birefringence
Indices of refraction
� Formula (Toll 1952)
� Two small parameters� Two small parameters
� dimensionless field strength (recall: )
� dimensionless probe frequency
� Note dependence on product
Experiment: measure ellipticity
Phase retardation of e+
Analysis
� ellipticity (squared)
� Power law suppressed…� Power law suppressed…
� Optimal scenario @ ELI
� large intensity:
� large probe frequency (X-ray, ):
� Still experimental challenge!
� Theory: for large , refractive indices develop imaginary part, (Toll 1952)
� Reason: pair creation (→ optical theorem)
New frontiers: increase parameters
� Q: can one get there?
� Large : currently science fiction
� Large : use Compton backscattering!
Parameter range
� : Compton backscattering off high-energy (from linac or wake field accelerator)
10-6 10-4 10-2 1
1
103
106
Vulcan
1 P
W
10
PW
ELI
all-optical
Backscattered(5 GeV )
SLAC
� for : 3 GeV @ ELI, 10 GeV @ Vulcan10PW
Large-ν birefringence I
Toll 1952 Shore 2007
� NB: SLAC E-144 had (K. Langfeld)
Large-ν birefringence II
� For , : find anomalous dispersion and
� Possibly, alternative signal for PP
� Subtle interplay between probe energy (ν) and laser intensity (ǫ2)
� Open questions:
� Polarimetry for high-energy γ’s ?
� Experimental signatures ?
5. Conclusion
Strong laser fields
� Direct laser acceleration in vacuum
� Proof of principle
� More advanced: plasma wake field acceleration
� Strong field QED� Strong field QED
� Absorptive: Pair creation – at which field strength?
� Dispersive: vacuum birefringence
� Scattering processes: no thresholds
� Gamma-gamma scattering
� High-intensity Compton