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